Tests for Sequences and Series

Nth Term

a_n

What does the nth term test for?

divergence

Nth Term diverges conditions

lim a ≠ 0

Geometric Series

arⁿ

Geometric Series Convergence Conditions

l r l < 1

Geometric Series Divergence Conditions

l r l ≥ 1

What does the Geometric Series converge to?

the sum; S = a / (1-r)

Telescoping Series

(a_n - a_n+1)

Telescoping Series Convergence Conditions

lim a_n = L; L has to be a finite number

What does the telescoping Series converge to?

the sum; S = a₁ - L

P-series test

1 / n^p

P-series test convergence conditions

p > 1

P-series test divergence conditions

0 < p ≤ 1

What if p = 1 in a p-series test?

divergent harmonic series

Alternating Series Test

(-1)ⁿ * a_n or (-1)ⁿ⁺¹ * a_n+1

Alternating Series Test convergent conditions

0 < a_n+1 ≤ a_n (nonincreasing) & lim a_n = 0

Alternating Series Test Remainder

l S - S_n l = l R_n l ≤ a_n+1

Integral Test conditions for f(x)

positive, continuous, and decreasing

Integral Test

a_n = f(n)

Integral Test Convergence Conditions

∫ f(x) dx converges, then series converges

Integral Test Divergence Conditions

∫ f(x) dx diverges, then series diverges

Ratio Test

a_n

Ratio Test Convergence Conditions

lim l a_n+1 / a_n l < 1

Ratio Test Divergence Conditions

lim l a_n+1 / a_n l > 1 or = ∞

Ratio Test is inconclusive if?

lim l a_n+1 / a_n l = 1

Root Test

a_n

Root Test Convergence Conditions

lim n√l a_n l < 1

Root Test Divergence Conditions

lim n√l a_n l > 1

The root test is inconclusive when?

lim n√l a_n l = 1

Direct Comparison Test

a_n

Direct Comparison Test Convergence Conditions

0 < a_n ≤ b_n and b_n converges

Direct Comparison Test Divergence Conditions

0 < b_n ≤ a_n and b_n diverges

Limit Comparison Test

a_n

Limit Comparison Test Convergence Conditions

lim (a_n / b_n) = L > 0 & b_n converges

Limit Comparison Test Divergence Conditions

lim (a_n / b_n) = L > 0 & b_n diverges

L must be what in limit comparison test?

finite and positive

Absolute vs. Conditional Convergence

once identified as convergence, use tests to see if l a_n l is convergent; yes- absolute or no- conditional

Sequence

a list; ex: 2, 4, 6, 8

Series

ex: 2 + 4 + 6 + 8

Recursive Definition (Implicit Definition) vs. Explicit Definition for Sequences

recursive- a_n+1 = a_n + x; explicit- a_n = 3n - 2