tests for convergence

Question 1

geometric series

Answer 1

if abs. value of r is ⟨ 1, converges
Sum of convergent series is (1st term)/1-r

Question 2

telescoping series

Answer 2

always converges, decompose series via partial fractions, and then cancel out terms until only the sum remains

Question 3

nth term test

Answer 3

if the limit as n approaches infinity does not equal 0, it will converge
*even if it does equal 0, does not mean it automatically converges

Question 4

integral test

Answer 4

1. f(x) is positive, continuous, decreasing
2. if the integral from k to infinity of f(x) converges, then sum of a sub n will converge
3. same if integral diverges, sum will diverge
   *to see if you should use this, see if you see a derivative for u-sub.

Question 5

p-series

Answer 5

∑from n=1 to infinity of 1/n ^p
if p ⟩ 1 –> converges
if p ≤1 –> diverges

Question 6

simple harmonic series

Answer 6

∑1/n –> always diverges

Question 7

alternating simple harmonic series

Answer 7

converges conditionally (needs the alternator)

Question 8

direct comparison test

Answer 8

1. 0 < a subn < b subn for all n
2. if ∑b subn converges, ∑a subn converges
3. if ∑a subn diverges, ∑b subn diverges

Question 9

limit comparison test

Answer 9

a subn > 0, b subn > 0
lim n goes to infinity of a/b = L
1. L is finite and positive
2. then ∑b subn + ∑a subn either both converge or both diverge

Question 10

alternating series test

Answer 10

1. needs an alternator
2. lim as n goes to infinity a subn has to go to 0
3. an+1 is always ≤ a of n (terms get smaller as series goes on)
   if all true, will converge!

Question 11

alternating series remainder

Answer 11

1. an+1 is always ≤ a of n
   [S-Sn]=[Rn] ≤ a of n+1

Question 12

maximum error in approximation is the last term you didn’t use

Question 13

absolute convergence

Answer 13

1. converges w/ or w/o the alternator
2. if ∑An abs. converges, ∑[An] converges
3. if absolute value converges, the series converges

Question 14

conditional convergence

Answer 14

1. converges only w/ alternator
2. ∑An condit. converges, ∑[An] diverges
3. the absolute value diverges, but the original converges

Question 15

Ratio Test

Answer 15

lim n goes to infinity of [(An+1)/An]
1. if limit is less than 1, series converges absolutely
2. if limit is greater than to 1, or goes to infinity, series diverges
3. inconclusive test if series = 1

Question 16

Root Test

Answer 16

n√[An]
1. if less than 1, converges
2. if greater than 1 or = to infinity, diverges
3. inconclusive if = 0
*same rules as ratio test