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equilibrium solution : any solution to the differential equation of the formy=c,y=c,whereccis a constant | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
Eulerâs Method : a numerical technique used to approximate solutions to an initial-value problem | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
general solution (or family of solutions) : the entire set of solutions to a given differential equation | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
growth rate : the constantr>0r>0in the exponential growth functionP(t)=P0ertP(t)=P0ert | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
initial population : the population at timet=0t=0 | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
initial value(s) : a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
initial velocity : the velocity at timet=0t=0 | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
initial-value problem : a differential equation together with an initial value or values | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
integrating factor : any functionf(x)f(x)that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
linear : description of a first-order differential equation that can be written in the forma(x)yâ²+b(x)y=c(x)a(x)yâ²+b(x)y=c(x) | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
logistic differential equation : a differential equation that incorporates the carrying capacityKKand growth raterrinto a population model | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
order of a differential equation : the highest order of any derivative of the unknown function that appears in the equation | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
particular solution : member of a family of solutions to a differential equation that satisfies a particular initial condition | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
phase line : a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
separable differential equation : any equation that can be written in the formyâ²=f(x)g(y)yâ²=f(x)g(y) | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
separation of variables : a method used to solve a separable differential equation | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
solution curve : a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
solution to a differential equation : a functiony=f(x)y=f(x)that satisfies a given differential equation | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
standard form : the form of a first-order linear differential equation obtained by writing the differential equation in the formyâ²+p(x)y=q(x)yâ²+p(x)y=q(x) | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
step size : the incrementhhthat is added to thexxvalue at each step in Eulerâs Method | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
threshold population : the minimum population that is necessary for a species to survive | https://openstax.org/books/calculus-volume-2/pages/4-key-terms |
To determine the convergence of a sequence given by an explicit formulaan=f(n),an=f(n),we use the properties of limits for functions. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
If{an}{an}and{bn}{bn}are convergent sequences that converge toAAandB,B,respectively, andccis any real number, then the sequence{can}{can}converges toc·A,c·A,the sequences{an±bn}{an±bn}converge toA±B,A±B,the sequence{an·bn}{an·bn}converges toA·B,A·B,and the sequence{an/bn}{an/bn}converges toA/B,A/B,providedBâ... | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
The geometric sequence{rn}{rn}converges if and only if|r|<1|r|<1orr=1.r=1. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Given the infinite seriesân=1âan=a1+a2+a3+â¯ân=1âan=a1+a2+a3+â¯and the corresponding sequence of partial sums{Sk}{Sk}whereSk=ân=1kan=a1+a2+a3+â¯+ak,Sk=ân=1kan=a1+a2+a3+â¯+ak,the series converges if and only if the sequence{Sk}{Sk}converges. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
The geometric seriesân=1âarnâ1ân=1âarnâ1converges if|r|<1|r|<1and diverges if|r|â¥1.|r|â¥1.For|r|<1,|r|<1,ân=1âarnâ1=a1âr.ân=1âarnâ1=a1âr. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
The harmonic seriesân=1â1n=1+12+13+â¯ân=1â1n=1+12+13+â¯diverges. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
A series of the formân=1â[bnâbn+1]=[b1âb2]+[b2âb3]+[b3âb4]+â¯+[bnâbn+1]+â¯ân=1â[bnâbn+1]=[b1âb2]+[b2âb3]+[b3âb4]+â¯+[bnâbn+1]+â¯is a telescoping series. Thekthkthpartial sum of this series is given bySk=b1âbk+1.Sk=b1âbk+1.The series will converge if and only iflimkââbk+1limkââ... | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Iflimnââanâ0,limnââanâ0,then the seriesân=1âanân=1âandiverges. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Iflimnââan=0,limnââan=0,the seriesân=1âanân=1âanmay converge or diverge. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Ifân=1âanân=1âanis a series with positive termsananandffis a continuous, decreasing function such thatf(n)=anf(n)=anfor all positive integersn,n,thenân=1âanandâ«1âf(x)dxân=1âanandâ«1âf(x)dxeither both converge or both diverge. Furthermore, ifân=1âanân=1âanconverges, then theNthNthpartial s... | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Thep-seriesân=1â1/npân=1â1/npconverges ifp>1p>1and diverges ifpâ¤1.pâ¤1. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
The comparison tests are used to determine convergence or divergence of series with positive terms. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
When using the comparison tests, a seriesân=1âanân=1âanis often compared to a geometric orp-series. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
For an alternating seriesân=1â(â1)n+1bn,ân=1â(â1)n+1bn,ifbk+1â¤bkbk+1â¤bkfor allkkandbkâ0bkâ0askââ,kââ,the alternating series converges. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
Ifân=1â|an|ân=1â|an|converges, thenân=1âanân=1âanconverges. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
For the ratio test, we considerÏ=limnââ|an+1an|.Ï=limnââ|an+1an|.IfÏ<1,Ï<1,the seriesân=1âanân=1âanconverges absolutely. IfÏ>1,Ï>1,the series diverges. IfÏ=1,Ï=1,the test does not provide any information. This test is useful for series whose terms involve factorials. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
For the root test, we considerÏ=limnââ|an|n.Ï=limnââ|an|n.IfÏ<1,Ï<1,the seriesân=1âanân=1âanconverges absolutely. IfÏ>1,Ï>1,the series diverges. IfÏ=1,Ï=1,the test does not provide any information. The root test is useful for series whose terms involve powers. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
For a series that is similar to a geometric series orpâseries,pâseries,consider one of the comparison tests. | https://openstax.org/books/calculus-volume-2/pages/5-key-concepts |
â n = 1 â 1 n = 1 + 1 2 + 1 3 + 1 4 + ⯠â n = 1 â 1 n = 1 + 1 2 + 1 3 + 1 4 + ⯠| https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
â n = 1 â a r n â 1 = a 1 â r for | r | < 1 â n = 1 â a r n â 1 = a 1 â r for | r | < 1 | https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
If a n â 0 as n â â , â n = 1 â a n diverges . If a n â 0 as n â â , â n = 1 â a n diverges . | https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
â n = 1 â 1 n p { converges if p > 1 diverges if p ⤠1 â n = 1 â 1 n p { converges if p > 1 diverges if p ⤠1 | https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
â« N + 1 â f ( x ) d x < R N < â« N â f ( x ) d x â« N + 1 â f ( x ) d x < R N < â« N â f ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
â n = 1 â ( â1 ) n + 1 b n = b 1 â b 2 + b 3 â b 4 + ⯠or â n = 1 â ( â1 ) n + 1 b n = b 1 â b 2 + b 3 â b 4 + ⯠or â n = 1 â ( â1 ) n b n = â b 1 + b 2 â b 3 + b 4 â ⯠â n = 1 â ( â1 ) n b n = â b 1 + b 2 â b 3 + b 4 â ⯠| https://openstax.org/books/calculus-volume-2/pages/5-key-equations |
absolute convergence : if the seriesân=1â|an|ân=1â|an|converges, the seriesân=1âanân=1âanis said to converge absolutely | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
alternating series : a series of the formân=1â(â1)n+1bnân=1â(â1)n+1bnorân=1â(â1)nbn,ân=1â(â1)nbn,wherebnâ¥0,bnâ¥0,is called an alternating series | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
alternating series test : for an alternating series of either form, ifbn+1â¤bnbn+1â¤bnfor all integersnâ¥1nâ¥1andbnâ0,bnâ0,then an alternating series converges | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
arithmetic sequence : a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
bounded above : a sequence{an}{an}is bounded above if there exists a constantMMsuch thatanâ¤Manâ¤Mfor all positive integersnn | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
bounded below : a sequence{an}{an}is bounded below if there exists a constantMMsuch thatMâ¤anMâ¤anfor all positive integersnn | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
bounded sequence : a sequence{an}{an}is bounded if there exists a constantMMsuch that|an|â¤M|an|â¤Mfor all positive integersnn | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
comparison test : if0â¤anâ¤bn0â¤anâ¤bnfor allnâ¥Nnâ¥Nandân=1âbnân=1âbnconverges, thenân=1âanân=1âanconverges; ifanâ¥bnâ¥0anâ¥bnâ¥0for allnâ¥Nnâ¥Nandân=1âbnân=1âbndiverges, thenân=1âanân=1âandiverges | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
conditional convergence : if the seriesân=1âanân=1âanconverges, but the seriesân=1â|an|ân=1â|an|diverges, the seriesân=1âanân=1âanis said to converge conditionally | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
convergence of a series : a series converges if the sequence of partial sums for that series converges | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
convergent sequence : a convergent sequence is a sequence{an}{an}for which there exists a real numberLLsuch thatananis arbitrarily close toLLas long asnnis sufficiently large | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
divergence of a series : a series diverges if the sequence of partial sums for that series diverges | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
divergence test : iflimnââanâ0,limnââanâ0,then the seriesân=1âanân=1âandiverges | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
divergent sequence : a sequence that is not convergent is divergent | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
explicit formula : a sequence may be defined by an explicit formula such thatan=f(n)an=f(n) | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
geometric sequence : a sequence{an}{an}in which the ratioan+1/anan+1/anis the same for all positive integersnnis called a geometric sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
geometric series : a geometric series is a series that can be written in the formân=1âarnâ1=a+ar+ar2+ar3+â¯ân=1âarnâ1=a+ar+ar2+ar3+⯠| https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
harmonic series : the harmonic series takes the formân=1â1n=1+12+13+â¯ân=1â1n=1+12+13+⯠| https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
index variable : the subscript used to define the terms in a sequence is called the index | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
infinite series : an infinite series is an expression of the forma1+a2+a3+â¯=ân=1âana1+a2+a3+â¯=ân=1âan | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
integral test : for a seriesân=1âanân=1âanwith positive termsan,an,if there exists a continuous, decreasing functionffsuch thatf(n)=anf(n)=anfor all positive integersn,n,thenân=1âanandâ«1âf(x)dxân=1âanandâ«1âf(x)dxeither both converge or both diverge | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
limit comparison test : supposean,bnâ¥0an,bnâ¥0for allnâ¥1.nâ¥1.Iflimnââan/bnâLâ0,limnââan/bnâLâ0,thenân=1âanân=1âanandân=1âbnân=1âbnboth converge or both diverge; iflimnââan/bnâ0limnââan/bnâ0andân=1âbnân=1âbnconverges, thenân=1âanân=1âanconverges. Iflimnâ... | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
limit of a sequence : the real numberLLto which a sequence converges is called the limit of the sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
monotone sequence : an increasing or decreasing sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
p-series : a series of the formân=1â1/npân=1â1/np | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
partial sum : thekthkthpartial sum of the infinite seriesân=1âanân=1âanis the finite sumSk=ân=1kan=a1+a2+a3+â¯+akSk=ân=1kan=a1+a2+a3+â¯+ak | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
ratio test : for a seriesân=1âanân=1âanwith nonzero terms, letÏ=limnââ|an+1/an|;Ï=limnââ|an+1/an|;if0â¤Ï<1,0â¤Ï<1,the series converges absolutely; ifÏ>1,Ï>1,the series diverges; ifÏ=1,Ï=1,the test is inconclusive | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
recurrence relation : a recurrence relation is a relationship in which a termananin a sequence is defined in terms of earlier terms in the sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
remainder estimate : for a seriesân=1âanân=1âanwith positive termsananand a continuous, decreasing functionffsuch thatf(n)=anf(n)=anfor all positive integersn,n,the remainderRN=ân=1âanâân=1NanRN=ân=1âanâân=1Nansatisfies the following estimate:â«N+1âf(x)dx<RN<â«Nâf(x)dxâ«N+1âf(x)dx<RN<â... | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
root test : for a seriesân=1âan,ân=1âan,letÏ=limnââ|an|n;Ï=limnââ|an|n;if0â¤Ï<1,0â¤Ï<1,the series converges absolutely; ifÏ>1,Ï>1,the series diverges; ifÏ=1,Ï=1,the test is inconclusive | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
sequence : an ordered list of numbers of the forma1,a2,a3,â¦a1,a2,a3,â¦is a sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
telescoping series : a telescoping series is one in which most of the terms cancel in each of the partial sums | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
term : the numberananin the sequence{an}{an}is called thenthnthterm of the sequence | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
unbounded sequence : a sequence that is not bounded is called unbounded | https://openstax.org/books/calculus-volume-2/pages/5-key-terms |
For a power series centered atx=a,x=a,one of the following three properties hold:The power series converges only atx=a.x=a.In this case, we say that the radius of convergence isR=0.R=0.The power series converges for all real numbersx. In this case, we say that the radius of convergence isR=â.R=â.There is a real num... | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The power series converges only atx=a.x=a.In this case, we say that the radius of convergence isR=0.R=0. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The power series converges for all real numbersx. In this case, we say that the radius of convergence isR=â.R=â. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
There is a real numberRsuch that the series converges for|xâa|<R|xâa|<Rand diverges for|xâa|>R.|xâa|>R.In this case, the radius of convergence isR. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
If a power series converges on a finite interval, the series may or may not converge at the endpoints. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The ratio test may often be used to determine the radius of convergence. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The geometric seriesân=0âxn=11âxân=0âxn=11âxfor|x|<1|x|<1allows us to represent certain functions using geometric series. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Given two power seriesân=0âcnxnân=0âcnxnandân=0âdnxnân=0âdnxnthat converge to functionsfandgon a common intervalI, the sum and difference of the two series converge tof±g,f±g,respectively, onI. In addition, for any real numberband integermâ¥0,mâ¥0,the seriesân=0âbxmcnxnân=0âbxmcnxnconverges ... | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Given two power series that converge on an interval(âR,R),(âR,R),the Cauchy product of the two power series converges on the interval(âR,R).(âR,R). | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Given a power series that converges to a functionfon an interval(âR,R),(âR,R),the series can be differentiated term-by-term and the resulting series converges tofâ²fâ²on(âR,R).(âR,R).The series can also be integrated term-by-term and the resulting series converges toâ«f(x)dxâ«f(x)dxon(âR,R).(âR,R). | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Taylor polynomials are used to approximate functions near a valuex=a.x=a.Maclaurin polynomials are Taylor polynomials atx=0.x=0. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Thenth degree Taylor polynomials for a functionffare the partial sums of the Taylor series forf.f. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
If a functionffhas a power series representation atx=a,x=a,then it is given by its Taylor series atx=a.x=a. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
A Taylor series forffconverges toffif and only iflimnââRn(x)=0limnââRn(x)=0whereRn(x)=f(x)âpn(x).Rn(x)=f(x)âpn(x). | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The Taylor series forex,sinx,sinx,andcosxcosxconverge to the respective functions for all realx. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
The binomial series is the Maclaurin series forf(x)=(1+x)r.f(x)=(1+x)r.It converges for|x|<1.|x|<1. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Power series can be used to solve differential equations. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
Taylor series can be used to help approximate integrals that cannot be evaluated by other means. | https://openstax.org/books/calculus-volume-2/pages/6-key-concepts |
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