Flashcards in Calculus II Final 9.3-9.6, 10.1-10.2 Deck (16):

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## Geometric Series

###
series of the form r^n

if |r|

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## Telescopic Series of the form 1/(n(n+1)

### = ((1/n)-(1/n-1)) converges to 1-1(n+1)

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## Telescopic Series of the form 1/(a^n)-1/(a^(n+1))

### converges to 1-1/(a^(n+1))

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## Divergence test

###
if the series convenes then the lim of the sequence=0

if the lim of the sequence is not=0 then the series diverges

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## Integral test

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if f is a continuous, positive, and decreasing function;

set an=f(n)

then the series an converges if the integral of f(n) is less than infinity

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## Ratio test

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ak is a series such that ak>0

r=lim(ak+1/ak)

if 01 then ak diverges

if r=1 then the ratio test is inconclusive

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## Absolute and conditional convergence

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1)a series ak is absolutely convergent if |ak| converges

2) if series ak converges but |ak| diverges then ak converges conditionally

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## limit comparison test

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find a bk to compare ak with

take the lim of ak/bk

if the lim= to a # then the series diverges

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## Comparison test

###
Ak is the series your evaluating, bk is the series you pick that is similar but easier to evaluate

if ak is less than bk and bk converges, then ak converges

if bk is less than ak and bk diverges, then ak diverges

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## Alternating series test

###
A series of the form (-1)^(k+1)*ak

if the limit as k->infinity of ak=0

then the series converges

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## What is a MacLaurin series?

### A taylor series in which a=0

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## Taylor Polinomial

### Pn(x)=f(a)+f'(a)*(x-a)+(f"(a)/2!)*(x-a)^2.....+(f^n(a)/n!)*(x-a)^n where f(x) is centered at a

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## Taylor series

### series (f^k(a)/k!)*(x-a)^k

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## Power series

### series Ck(x-a)^k where Ck are the coefficients of the power series and the function is centered at a

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## How to find the radius of convergence(R)

### R=lim as k->infinity of 1/|(Ck+1)/(Ck)|

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