INFINITE SERIES: Convergence/Divergence Tests

The Sandwich Theorem for Sequences:

If an ≤ bn ≤ cn for all n beyond some index N, and if lim an = lim cn = L, then lim bn = L also.

The nth Term Test for Divergence: ∞

If lim a n ≠ 0, or if lim a n fails to exist, then an diverges. n→∞ n→∞ ∑ n=1

The Non-Decreasing Sequence Theorem:

A non-decreasing sequence converges if and only if its terms are bounded from above. If all the terms are less than or equal to M, then the limit of the sequence is less than or equal to M as well.

Comparison Test for Series of Non-negative Terms:

Let∑ an be a series with no negative terms :

1. Test for convergence:

The series ∑ a n converges if there is a convergent series ∑ c n with an ≤ cn

for all n > no , for some positive integer n o.

2. Test for divergence :

The series ∑ an diverges if there is a divergent series of non - negative terms d n with

an ≥ d n for all n > no .

Integral Test:

Let a n = f(n) where f(x) is a continuous, positive, decreasing function of x for all x ≥ 1. ∞ Then the series a and the integral f(x)dx both converge or both diverge. ∑ n ∫ 1

Limit Comparison Test: 1. Test for convergence :

If a n ≥ 0 for n > no and there is a convergent series ∑ cn such that cn > 0

a n and lim < ∞ then ∑ a n converges. cn

2. Test for divergence :

If a n ≥ 0 for n ≥ no and there is a divergent series ∑ dn such that dn > 0 and

a n lim > 0, then ∑ a n diverges. dn

Simplified Limit Comparison Test:

If the terms of the two series ∑ an and ∑ bn are positive for n ≥ no , and the limit of a n is finite and positive, then both series converge or both series diverge. bn

The Ratio Test:

a n +1 Let a n be a series with positive terms, and suppose that lim = p ∑ n→∞ a n Then (a) the series converges if p < 1 (b) the series diverges if p > 1 (c) the series may converge or it may diverge if p = 1 (The test provides no informatio n.)

n Let ∑ a n be a series with a n ≥ 0 for n ≥ n 0 , and suppose that a n → p Then (a) the series converges if p < 1 (b) the series diverges if p > 1 (c) the test is not conclusive if p = 1

th The n - Root Test:

Limit of the nth term of a convergent series. ∞ If ∑an converges, then an → 0. n=1 nth-term test for divergence: ∞ ∑an diverges if lim an fails to exist or is different from zero. n=1 n→∞

Examples: ∞ 1. ∑ n 2 diverges because n 2 → ∞ n=1 ∞ n +1 n +1 2. ∑ diverges because → 1 n=1 n n ∞ 3. ∑(-1)n+1 diverges because lim(−1) n+1 does not exist n=1 n→∞ ∞ − n ⎛ − n ⎞ −1 4. ∑ diverges because lim⎜ ⎟ = ≠ 0 n=1 2n + 5 n→∞⎝ 2n + 5 ⎠ 2

The Absolute Convergence Theorem:

∞ ∞ If ∑ a n converges, then ∑a n converges. n>1 n=1

2 How to Test a Power Series for Convergence:

1. Use the Ratio Test (or nth -Root Test) to find the interval where the series converges absolutely.

2. If the interval of absolute convergence is finite, test for convergence or divergence at each of the two endpoints. Use a Comparison Test, the Integral Test, or the Alternating Series Theorem, not the Ratio Test nor the nth –Root Test.

3. If the interval of absolute converge is a - h < x < a + h, the series diverges (it does not even converge conditionally) for x - a > h because for those values of x, the nth term does not approach zero.

Revised: Spring 2005 STUDENT LEARNING ASSISTANCE CENTER (SLAC) Texas State University-San Marcos 3