MATH 1262 Calculus II - Chapter 11 Key Terms and Review

Sequence

an ordered list of numbers, (an) or {an}

Series

the sum of terms in a sequence, sigma an

Infinite Series Theorem

If sigma n=1 to infinity an converges, then lim n to infinity of an = 0

Test for Divergence (TFD)

If lim (n to infinity) an does not equal 0, then the series of an diverges

Harmonic Series

{1/n}, always diverges

Geometric Series

Sum of ar^n-1, becomes Sn = a/1 - r (1 - r^n)

If | r | < 1

Convergent by Geometric Series Test

if | r | is greater than or equal to 1

Divergent by Geometric Series Test

Telescopic Series

Series where most terms cancel out, allows for direct evaluation of lim (n to infinity) Sn.

Integral Test

For {an} that is positive and decreasing (an > an+1), the series of an is the sum of the "areas"

If the integral from 1 to infinity f(x)dx converges, then…

the series an also converges

If the integral from 1 to infinity f(x)dx diverges, then…

the series an also diverges

The series (an) converges if and only if…

the integral from 1 to infinity of f(x)dx converges (cts analog)

P-Series

series of 1/n^p, converges for p > 1, diverges for 0 < p <= 1

Direct Comparison Test

Compares series to known convergent/divergent series.

If an

the series an also converges

If bn

the series bn also diverges

Limit Comparison Test

if lim as n approaches ∞ of ratio of comparison series/general term is positive and finite, then series behaves like comparison series

If 0 < L < infinity, then…

either sum of an & sum of bn both converge or diverge (same behavior)

if L = 0 & sum of bn converges, then…

sum of an also converges

If L = infinity & sum of bn diverges, then…

sum of an also diverges

Alternating Series

The terms of a series change between positive and negative

Alternating Series Test (AST)

A series (-1)^n-1 bn converges if bn > 0, bn decreases, and lim (n to infinity) bn = 0

Absolute Convergence

If the series of | an | is convergent, then the sum of an is absolutely convergent

Ratio Test

For series of an, compute L = lim | (an+1) / (an) |. It converges if L < 1, is inconclusive if L = 1, and diverges if L > 1

Factorials (a!)

A number multiplied by all numbers leading up to it (eg. 5! = 5 x 4 x 3 x 2 x 1). Can be split into smaller segments (eg 5! = 5 x 4!) to help with cancellations

0! = ?

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