Infinite Series Flashcards
Define series
A series is a summing up of terms
Define partial sum
SN = sum(N,n=0)an
Series sum(inf,n=0)x^n converges?
To 1/(1-x) iff |x| less 1
Integral test
Suppose f is a cts, pos, dec function on [1, inf] and an=f(n) Then If S(inf,1)f(x)dx converges, then sum(inf,n=1)an converges If S(inf,1)f(x)dx diverges, then sum(inf,n=1)an diverges
nth term test
For sum(inf,n=1)an, if lim(n to inf)an == 0, then the series does not converge
p series
Sum(inf,n=1)1/not converges iff p great 1
Remainder estimate for integral test
Suppose f(n) = an, where f is cts, pos, dec If sum(inf,n=1)an is convergent, then S(inf,N+1)f(x)dx lessequal RN lessequal S(inf,N)f(x)dx
What is the remainder RN
Sum(inf,n=1)an-SN
Error in approximation
Euler-Maclaurin Summation
Sum(n,k=m)f(k) = S(n,m)f(x)dx + 1/2[f(m) + f(n)] + 1/12[f’(m) - f’(n)] + error
Comparison tests - series
Given the series sum(inf,0)an and sum(inf,0)bn with 0 lessequal an lessequal bn
Then
If sumbn converges, then suman diverges
If suman diverges, then sumbn diverges
Limit comparison test - series
Given the series suman and sumbn with positive terms (an,bn greatequal 0)
If lim(n to inf)an/bn = ρ, with 0 less ρ less inf
Then both series either converge or diverge
Alternating series test
If lim(n to inf)Pn = 0 and P(n+1) lessequal Pn, then sum(inf,n=0)(-1)^nPn converges
Alternating series remainder
error RN = |sum(inf,n=0)(-1)^nPn - sum(N,n=0)(-1)^nPn| lessequal P(N+1)
Define absolute convergence
A series suman is called absolutely convergent if sum|an| converges
Define conditionally convergent
A series suman is called conditionally convergent if it is convergent but not absolutely
Ratio test
Given a series suman, take the limit L=lim(n to inf)|a(n+1)/an| L less 1 absolutely convergent L great 1 divergent L = 1 inconclusive
Root test
Given a series suman, take the limit L = lim(n to inf)nroot(|an|) L less 1 absolutely convergent L great 1 divergent L = 1 inconclusive
Define power series
A power series is an infinite series involving powers of a variable
Sum(inf,n=0)Cn(x)^n or sum(inf,n=0)Cn(x-x0)^n
Define interval/radius of convergence
Interval of convergence is x values where series converges
Radius of convergence is…who knows
Why does absolute convergence matter
Absolutely convergent series behave just like ordinary functions
Series flow chart
1) does lim(n to inf)an = 0
2) is series alternating
3) ratio test
4) is it a p series
5) is it closely related to |x|^n
6) cts, pos, dec, easy to integrate
Euler-Maclaurin Error
|error| lessequal 1/120[S(n,m)|d3f/dx3|dx
