Series Convergence/Divergence Tests

nth Term Test for Divergence

• If lim a_n as n→infinity ≠ 0 or does not exist, then the series diverges.

• If lim a_n as n→infinity = 0, then the test is inconclusive.

Geometric Series Test

The geometric series: sum of ar^n converges to Sum = a/(1-r) if |r|<1 and diverges |r|>=1 and a≠0

P-series test: general

The p-series: sum of 1/n^p, with n starting at 1, converges if p>1 and diverges if p<=1.

P-series test: harmonic series

The harmonic series: sum of 1/n, with n starting at 1, diverges

P-series test: alternating harmonic series

The alternating harmonic series: sum of ((-1)^(n+1))/n, with n starting at 1, converges.

Ratio Test: |a_(n+1)/a_n|<1

Let: sum of a_n be a series with nonzero terms.

• If lim as n→infinity |a_(n+1)/a_n|<1, then the series converges absolutely.

Ratio Test: lim n→infinity |a_(n+1)/a_n|>1

Let: sum of a_n be a series with nonzero terms.

• If lim n→infinity |a_(n+1)/a_n|>1, then the series is divergent.

Ratio Test: lim n→infinity |a_(n+1)/a_n|=1

Let: sum of a_n be a series with nonzero terms.

• If lim n→infinity |a_(n+1)/a_n|=1, then the test is inconclusive.

Direct Comparison Test: sum of b_n, with n starting at 1, converges

Let 0<=a_n<=b_n, for all sufficiently large n.

• If the sum of b_n, with n starting at 1, converges, then the sum of a_n, with n starting at 1, also converges.

Direct Comparison Test: sum of a_n, with n starting at 1, diverges

Let 0<=a_n<=b_n, for all sufficiently large n.

• If the sum of a_n, with n starting at 1, diverges, then the sum of b_n, with n starting at 1, also diverges.

Limit Comparison Test

Let sum of a_n, with n starting at 1, and b_n, with n starting at 1, be a series of nonnegative terms, with a_n≠0 for all sufficiently large n.

And suppose that lim as n→infinity of b_n/a_n = c>0, where c is a finite number.

• Then the two series either both converge or both diverge.

Integral Test

f(x) must be positive, continuous, and decreasing on [1,infinity). Let a_n = f(n)

• Sum of a_n, with n starting at 1, converges if the improper integral of 1 to infinity of f(x)dx converges.

• If the improper integral diverges, then the infinite series diverges.

Alternating Series Test

Let sum of a_n, with n starting at 1, be a series such that:

• series alternating,

• |a_(n+1)|<=|a_n| for all n,

• and lim as n approaches infinity of a_n = 0.

Then the series converges.

What does sum of a_n, with n starting at 1, do if sum of |a_n|, with n starting at 1, converges.

sum of a_n, with n starting at 1, also converges.

When is a series, sum of a_n, absolutely convergent?

If the series, sum of |a_n|, converges.

When is a series, sum of a_n, conditionally convergent?

If the series, sum of a_n, converges but sum of |a_n| does not converge.