instruction stringlengths 12 30k |
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Over at [this MathOverflow question](http://mathoverflow.net/questions/12342/reductio-ad-absurdum-or-the-contrapositive/12400#12400), I posted the following answer to a similar question (and there are several other interesting answers there):
- *With good reason*, we mathematicians prefer a direct proof of an impl... |
I want to try to emulate what this application can do:
![alt text][1]
Given the red round dots (from the mouse) it is able to solve for the bezier handles given that the tension of the curve is set to 0.6. How could I solve for the bezier handles? Is there some way I could do this with the slope or perpendicular ... |
This is rather scheme-y, but there's a really nice paper by [Kempf][1] (hopefully you have institutional access :() that gives a very basic and elementary proof that the higher cohomology of a quasi-coherent sheaf on an affine scheme is trivial. The first part of the paper uses nothing more than the basic properties ... |
In DotA, there is a character called "[Axe](http://www.dotastrategy.com/hero-71-MogulKhanAxe.html)". Every time he is attacked, he has a chance to spin (17%) his blade and deal damage based on what level the skill is, 100 / 125 / 150 / 175 damage for levels 1, 2, 3, and 4. When the spin activates, it triggers the coold... |
If we have a curve $C:\\{ P(x,y) = 0 \\}$ and define $\omega=\frac{\mathrm{d}x}{y}$ then is
$$\int_0^A \omega + \int_0^B \omega = \int_0^{A \oplus B} \omega$$
(with $\oplus$ being addition on a group on the curve) a theorem?
I'm quite sure this is what's happening when C is an elliptic curve, but I have fail... |
Here is the simplest way I know. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax^2 + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From that, just multiply it all out and ... |
Wikipedia [sez][1]:
> The natural logarithm of x is often written "ln(x)", instead of log_e(x) especially in disciplines where it isn't written "log(x)". However, some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish ln notation," which he... |
How did the notation "ln" for "log base e" become so pervasive? |
Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax^2 + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From that, just ... |
You might look at Hartshorne's
[*Geometry: Euclid and Beyond*][1].
[1]: http://books.google.co.uk/books?id=EJCSL9S6la0C&lpg=PP1&dq=hartshorne&pg=PP1#v=onepage&q&f=false |
It is probably at least in part an instance of consumer lock-in, the phenomenon where the advantages of the majority of the market using the same or compatible products outweigh sufficiently minor differences of one version of the product versus another. (When I learned this term, the canonical example was VHS versus ... |
I came across the statement, that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations.
How exactly does generalizes Yoneda-lemma Cayley`s theorem? Can Cayley's theorem be deduced from Yoneda lemma, is it a generalization of a special case of Yo... |
Yoneda-Lemma as generalization of Cayley`s theorem? |
Denote by $D(a_1,\dots,a_n)$ the product $\prod_{j>i}(a_j-a_i)$. Assuming that $a_i$ are integers s.t. $a_1\le a_2\le\dots\le a_n$, proove that $D(a_1,...,a_n)/D(1,...,n)$ is the number of _Gelfand-Zetlin triangles_ (that is, [triangles](http://imrn.oxfordjournals.org/content/vol2010/issue13/images/medium/rnp223ueq3.gi... |
In group cohomology theory, we know that the cohomological dimension cd(G) for a profinite group is a fundamental numerical invariant. We say G is of virtual cohomological dimension n (denote it by vcd(G)) if there exists an open subgroup H such that cd(H)=n.
It seems that vcd(G) is not always a fixed integer and m... |
When does the virtual cohomological dimension become a numerical invariant? |
Cajori, in his *History of mathematical notations*, Vol. II, as far as I can see, mentions the notation «$\operatorname{ln}$» once when he is talking about notations for logarithm. He refers to [Irving Stringham, *Uniplanar Algebra*, (San Francisco 1893), p. xiii] as someone who used that notation; the way this guy is ... |
Cajori, in his *History of mathematical notations*, Vol. II, as far as I can see, mentions the notation «$\operatorname{ln}$» once when he is talking about notations for logarithm. He refers to [Irving Stringham, *Uniplanar Algebra*, (San Francisco 1893), p. xiii] as someone who used that notation; the way this guy is ... |
It is probably at least in part an instance of consumer lock-in, the phenomenon where the advantages of the majority of the market using the same or compatible products outweigh sufficiently minor differences of one version of the product versus another. (When I learned this term, the canonical example was VHS versus ... |
Does set ℝ^+ include zero? |
Just as Cayley's theorem states that every group is a subgroup of a symmetric group, Yoneda's lemma states that every (locally small) category $C$ embeds into a category of functors defined on $C$.
Specifically, Yoneda's lemma states that if $F:C\to Set$ is an arbitrary set-valued functor, then $F(A) = Nat(\hom(A,-)... |
Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax<sup>2</sup> + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From t... |
Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax<sup>2</sup> + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From t... |
Just as Cayley's theorem states that every group is a subgroup of a symmetric group, Yoneda's lemma states that every (locally small) category $C$ embeds into a category of functors defined on $C$.
Specifically, Yoneda's lemma states that if $F:C\to Set$ is an arbitrary set-valued functor, then $F(A) = Nat(\hom(A,-)... |
I've been experimenting with the summation of polynomials. My line of attack is to treat the subject the way I would for calculus, but not using limits.
By way of a very simple example, suppose I wish to add the all numbers between 10 and 20 inclusive, and find a polynomial which I can plug the numbers into to get m... |
Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax<sup>2</sup> + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From t... |
Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.
if f(x) = ax<sup>2</sup> + bx + c
Then the symmetry around x = k means that, for any h
f(k+h) = f(k-h)
From t... |
Does set $\mathbf ℝ^+$ include zero? |
Does set $\mathbb{R}^+$ include zero? |
$ 7 | 3^6 - 1,\; 11 | 3^{10} - 1 => 77 | 3^{120} - 1 => 77 | 3^{123} - 3^3 $. . . QED
While this case was too trivial to require such, you may be interested in looking at various generalizations of the Fermat-Euler theorem due to Carmichael. See my [post \[1\]] [1] for a couple of these, e.g.
**THEOREM 1** For... |
$ 7 | 3^6 - 1,\; 11 | 3^{10} - 1 => 77 | 3^{120} - 1 => 77 | 3^{123} - 3^3 $. . . **QED**
While this case was too trivial to require such, you may be interested in looking at various generalizations of the Fermat-Euler theorem due to Carmichael. See my [post \[1\]] [1] for a couple of these, e.g.
**THEOREM 1** ... |
Related to [this question][1].
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $k$ **sections** by $k$ axis-aligned hyperplanes in $\mathbb{R}^{n-1}$. For e.g., if $k=2$, we have $2$ $n$-hemispheres.
Here's the **question**: how do I find the center o... |
Related to [this question][1].
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $k$ **sections** by $k$ (note, $k<=n$) axis-aligned hyperplanes in $\mathbb{R}^{n-1}$. For e.g., if $k=2$, we have $2$ $n$-hemispheres.
Here's the **question**: how do I fi... |
Related to [this question][1].
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $k$ orthants by $k$ (note, $k<=n$) axis-aligned hyperplanes in $\mathbb{R}^{n-1}$. For e.g., if $k=2$, we have $2$ $n$-hemispheres.
Here's the **question**: how do I find t... |
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to start with the definition of one, yet many "official" definitions I have found too convoluted (or poorly written) for genera... |
How do you define functions for non-mathematicians? |
The way you've restated the definition is fairly common in contemporary high school books in the U.S. (perhaps changing "links between two sets" to "ordered pairs"). What I've seen a lot of in middle school and earlier algebra settings is the idea of a "function machine." The function machine graphic below is from [F... |
I suggest that everybody from now on uses ln, lg, and lb respectively for the natural, decimal, and binary logarithmic functions, reserving log for when the base is displayed explicitly or is some fixed arbitrary number if the scale doesn't matter. I do (with explanation if necessary at the first instance) . What could... |
Let $f$ be a function defined on an open interval $(a,b)$ which has continuous derivatives of order $1,2, \dots , n-1$, and there is a point $c \in (a,b)$ such that $f(c), f^{{1}}(c), \dots , f^{(n-1)}(c)$ are all $0$. However $f^{n}(c)$ is not $0$ and we can assume that $f^{n}(x) > 0$ for $x \in (a,b)$, however **no ... |
Related to [this question][1].
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $k$ orthants by $k$ (note, $k<=n$) axis-aligned hyperplanes in $\mathbb{R}^{n-1}$ passing through $\bf 0$. For e.g., if $k=2$, we have $2$ $n$-hemispheres.
Here's the **que... |
Related to [this question][1].
Note that I'm using the [geometer definition][2] of an $n$-sphere, i.e., the surface of a ball in $\mathbb{R}^n$
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperplanes (note, $k \l... |
Related to [this question][1].
Note that I'm using the [geometer definition][2] of an $n$-sphere of radius $r$, i.e.$ \{ x \in \mathbb{R}^n : \|x\|_2 = r \} $
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperpla... |
Related to [this question][1].
Note that I'm using the [geometer definition][2] of an $n$-sphere of radius $r$, i.e.$ \\{ x \in \mathbb{R}^n : \|x\|_2 = r \\} $
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperp... |
Classes are larger than sets.
What's larger than classes ? |
Sets. Classes. ... ? |
For fun, I like to liven-up the "black box"/machine view of a function by putting a monkey into the box. (I got pretty good at chalkboard-sketching a monkey that looked a little bit like [Curious George][1], but with a tail.)
Give the Function Monkey an input and he'll cheerfully give you an output. The Function Mon... |
Not a math student, so forgive me if the question seems trivial or if I pose it "wrong". Here goes...
Say I'm flipping a coin a n times. I am not sure if it's a "fair" coin, meaning I am not sure if it will come up heads 50% of the time and tail 50% of the time. Now, if after n throws it has come up heads exactly as... |
How to determine if coin comes up heads more often than tails? |
What are the best algebraic geometry textbooks for undergraduate students? |
How do I figure out what kind of distribution this is? |
See the MathOverflow question about exactly this: [Best Algebraic Geometry text book?](http://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne)
(Personally, from what little I've seen, [*Ideals, Varieties and Algorithms*](http://www.cs.amherst.edu/~dac/iva.html) by Cox, Litle a... |
What is the probability that exactly $i$ heads will result from $n$ coin tosses? |
See the MathOverflow question about exactly this: [Best Algebraic Geometry text book?](http://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne)
Personally, from what little I've seen, [*Ideals, Varieties and Algorithms*](http://www.cs.amherst.edu/~dac/iva.html) by Cox, Litle an... |
Given your prefatory comment, I'm going to avoid talking about the normal curve and the associated variables and use as much straight probability as possible.
Let's do a side problem first. If on a A-D multiple choice test you guess randomly, what's the probability you get 8 out of 10 questions right?
Each proble... |
Given your prefatory comment, I'm going to avoid talking about the normal curve and the associated variables and use as much straight probability as possible.
Let's do a side problem first. If on a A-D multiple choice test you guess randomly, what's the probability you get 8 out of 10 questions right?
Each proble... |
Can you divide a disk into 7 pieces of equal area, with 3 line segments? (You can surely divide it into 7 pieces, but could those have equal areas?)
(I can see with some visual arguments that the answer should be *no*, but I couldn't find a nice way to write it down.)
|
Everyone is linking to that MO question, so allow me to link to another MO [question][1] which is more relevant to this specific request.
Also, I should mention that it partially depends on what you are looking for. Should this be for self study with little help from professors? Then your best bet would be to follow... |
Set theory is completely full of new axioms, some of them expressing fundamental set principles, and many of them having consequences even in natural number arithmetic that are not provable without them (via consistency strength statements).
In the past fifty years of research in set theory, a major lesson that has ... |
Can you divide a disk into 7 pieces of equal area, with 3 line segments? (You can surely divide it into 7 pieces, but could those have equal areas?)
(This question was left unanswered at another forum. I can see with some visual arguments that the answer should be *no*, but I couldn't find a nice way to write it d... |
The point $(x,y)=\left( 0,0\right) $ is the minimum (maximum) if $a>0$ ($a<0$) of the parabola given by $y=ax^{2}$.
By making the change of coordinates $X=x+b/(2a),Y=y+(b^{2}-4ac)/(4a)$, equation $y=ax^{2}+bx+c$ is transformed into $Y=aX^{2}$, whose minimum ($a>0$) or maximum ($a<0$) occurs at the point $(X,Y)=(0,0)$,... |
Suppose the following series converges
$\sum_{n=1}^{\infty} a_{n}^{3}$
Does this imply that
$\sum_{n=1}^{\infty} \frac{a_{n}}{n}$
converges?
I was able to prove that the second series also converges by using the limit comparision test. Is there another way to show the second series converges (e.g. roo... |
The point $(x,y)=\left( 0,0\right) $ is the minimum (maximum) if $a>0$ ($a<0$) of the parabola given by $y=ax^{2}$.
By making the change of coordinates $X=x+b/(2a),Y=y+(b^{2}-4ac)/(4a)$, equation $y=ax^{2}+bx+c$ is transformed into $Y=aX^{2}$, whose minimum ($a>0$) or maximum ($a<0$) occurs at the point $(X,Y)=(0,0)$,... |
This one has been bugging me for a while - I have a number of definitions for a symmetric group, none of which I understand. Clearly, $S_n$ is a group, which I follow. Now, the wikipedia definition (most readily to hand) is:
> The symmetric group on a set X is the
> group whose underlying set is the
> collection o... |
What is a symmetric group? |
[This question][1] just reminded me of a conundrum I posed myself in my first year of university. I never did get a satisfactory answer...
>Let $a_n$ be a null sequence. Does it follow that $\sum \frac{a_n}{n}$ converges?
Any ideas?
[1]: http://math.stackexchange.com/questions/1766/convergent-series |
In the early days of commutative algebra people did not use the language of localization, so probably such proof exists. In particular, I have been told Kaplansky's early book (around 1950?) might be such reference, but I don't have it right now.
In any case, [Exercise 2.3](http://books.google.com/books?id=Fm_yPgZ... |
**Edit:** Bill Dubuque has pointed out that this answer is essentially a mathematical urban legend, albeit a pervasive one. I cannot delete an accepted answer, so here is a [link][1] to an answer of his on MO explaining it.
There's some of the history here in Bourbaki's _Commutative Algebra,_ in the appendix. Bas... |
[This question][1] just reminded me of a conundrum I posed myself in my first year of university. I never did get a satisfactory answer...
>Let $a_n$ be a [null sequence](http://www.answers.com/topic/null-sequence). Does it follow that $\sum \frac{a_n}{n}$ converges?
Any ideas?
[1]: http://math.stackexchan... |
Suppose $a_{n}>0$ and the following series converges
$\sum_{n=1}^{\infty} a_{n}^{3}$
Does this imply that
$\sum_{n=1}^{\infty} \frac{a_{n}}{n}$
converges?
I was able to prove that the second series also converges by using the limit comparision test. Is there another way to show the second series conve... |
If $\sum_{n=1}^{\infty} a_{n}^{3}$ converges does $\sum_{n=1}^{\infty} \frac{a_{n}}{n}$ converge? |
Motivation: A friend asked me this question.
The Problem: Suppose you start off with a dollar. You flip a fair coin, if it lands on heads you win 50 cents otherwise you lose 50 cents. If after n flips you have a nonzero amount of money, you win. Whats the probability you win? What about the limiting case as n tends ... |
This is the equation I'm having trouble with:
$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$
That's the non-vector form of the [universal law of gravitation][1] on the left and [Newton's second law of motion](http://en.wikipedia.org/wiki/Newton's_laws_of_motion) on the right. I assume that upon correctly modeling and... |
Can this gravitational field differential equation be solved, or does it not show what I intended? |
I just finished reading the Wikipedia article on the [Cauchy condensation test][1]. I understand the trapezoidal view, but apparently "the 'condensation' of terms is analogous to a substitution of an exponential function". Can anyone explain what is meant by this?
[1]: http://en.wikipedia.org/wiki/Cauchy%27s_conde... |
Not a math student, so forgive me if the question seems trivial or if I pose it "wrong". Here goes...
Say I'm flipping a coin a _n_ times. I am not sure if it's a "fair" coin, meaning I am not sure if it will come up heads and tails each with a propability of exactly 0.5. Now, if after _n_ throws it has come up head... |
Motivation: A friend asked me this question.
The Problem: Suppose you start off with a dollar. You flip a fair coin, if it lands on heads you win 50 cents otherwise you lose 50 cents. If after n flips you have a nonzero amount of money, you win. Whats the probability you win? What about the limiting case as n tends ... |
Can anyone explain the syntax on the following Wikipedia article on the [convergence of random variables][1]?
![alt text][2]
[1]: http://en.wikipedia.org/wiki/Convergence_of_random_variables
[2]: http://upload.wikimedia.org/math/1/b/d/1bd88f4785429841e1d24c2996cbb6e0.png |
Can anyone explain the syntax from the following Wikipedia article on the [convergence of random variables][1]?
![alt text][2]
In particular:
1. What is the squiggly line?
2. What are $d$, $D$, $L$, $L_X$?
3. Why does it say $X_n$ implies X? They don't seem to be statements
[1]: http://en.wikipedia.o... |
For a given $k>0$ constant, assuming that $x,y>0$. Also this equation has a particular name, or some mathematician associated with it?
|
How to solve the equation $x^y=y^x=k$? |
Let $A$, $X$, and $Y$ be arbitrary topological spaces. Let $F:X\times A\rightarrow Y$ be a continuous function. Let $P$ be the map from $X\times A$ to $Y\times A$ taking $(x,a)$ to $(F(x,a),a)$. Does it follow from continuity of $F$ that $P$ is continuous? |
Is this function continuous? |
I don't have enough rep to comment yet, so I'll make this CW since it's not a complete answer, just a remark.
Would it overcome some of the difficulty by using some sort of graph representation of the subdivided rectangle? Each rectangle is a vertex and an edge represents sharing a side, i.e.
the first rectangle ab... |
I'm mainly a CS major, but I also want to learn more advanced mathematics. I see lot of cross over between CS and applied math, same can't be said about pure math.
Definition of 'advanced': Something a pure math major learns during and after late undergraduate years.
What are the fields that combines advanced ma... |
**Edit:** Bill Dubuque has pointed out that much of this answer (specifically, the part about FLT) is essentially a mathematical urban legend, albeit a pervasive one. I cannot delete an accepted answer, so here is a [link][1] to an answer of his on MO explaining it.
Here is also a link to a related [question][2].
... |
Are there any books or online notes that cover the basics of lie groups using nonstandard analysis? Another thing I would like is a to see these things set in category theory (along the lines of *Algebra: Chapter 0* except for differential geometry?) |
Books on Lie Groups via nonstandard analysis? |
Let $X$ be a compact metric space. Given a sequence $x_n \in X$ and an ultrafilter $\mathcal{U}$ on $\mathbb{N}$, we can define the "Banach limit" of $\{x_n\}$ with respect to $\mathcal{U}$. This limit is the unique element of $X$ such that every neighborhood of it contains a $\mathcal{U}$-large collection of the sequ... |
Here is a non-answer which may prove interesting and tractable.
Given a configuration of rectangles that properly tile a surrounding rectangle, one can
represent the tiling in a number of ways. Two that come to mind I will call C and LL.
C represents each rectangle by giving its dimensions (x-span and y-span) and... |
What is the difference between convergence in distribution, convergence in probability and almost sure convergence?
**References**
[Convergence of random variables][1]
[1]: http://en.wikipedia.org/wiki/Convergence_of_random_variables |
I have just started auditing Linear Algebra, and the first thing we learned was how to solve a system of linear equations by reducing it to row echelon form (or reduced row echelon form). I know how to solve a system of linear equations using determinants, and so the using row echelon form seems to be very inefficient ... |
Is reducing a matrix to row echelon form useful at all? |
By little Fermat, $ 6,10|120 \Rightarrow 3^{120} \equiv 1 \pmod{7, 11} \Rightarrow 3^{123} \equiv 3^3 \pmod{77}$
More generally see these Fermat-Euler-Carmichael results from my [post \[1\]] [1]
**THEOREM 1**. For natural numbers $ a,e,n $ with $ e,n>1 $
$\; n|a^e-a$ for all a $\iff n$ is squarefree and prim... |
I am working a bit on a collection of Linear Algebra examples,
as well as some examples on induction. This is what is taught freshman year at our university.
I intend to release this to the public, either by selling printed copies or releasing it online.
Since I do not have experience using such material mysel... |
1. I think the only *real* utility of Cramer's rule -besides resolving 2x2 systems by hand- is that, for a general system of linear equations $AX = B$ with $\det A \neq 0$, it shows blatantly http://en.wikipedia.org/wiki/Cramer%27s_rule the continuous dependence of the solution with respect both $A$ and $B$.
2. But,... |
I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction, a... |
Why is Gimbal Lock an issue? |
I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction, a... |
Let $a_n\geq 0$ be a sequence of non-negative numbers. Consider the following two statements:
$$
\text{(I)}\qquad\qquad \lim_{n\to\infty} \frac{1}{n^2}\sum_{i=1}^n a_i =0,
$$
$$
\text{(II)}\qquad\qquad\qquad \sum_{n=1}^\infty \frac{a_n}{n^2}<\infty.
$$
>**Questions:** Does (I) imply (II)? Does (II) imply (I)? Ot... |
If $\sum_{n=1}^\infty \frac{a_n}{n^2}$ is finite, does $\frac{1}{n^2}\sum_{i=1}^n a_i$ go to $0$? |
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