Positive Term Infinite Series (continued)

In a Nutshell: One of the most straight forward tests for convergence of any infinite series

(positive term or alternating term) is the Nth term divergence test. It is quick and easy to

apply but doesn’t always provide information. The table below details this test.

Nth term divergence test

If lim a_n ≠ 0 or the limit does not exist, then ∑ a_n diverges.

n → ∞

Note: This test yields NO INFORMATION if the limit equals zero.

Another useful test restricted to positive term series is the integral test.

The sole purpose of the integral test is to see if ∑ a_n converges. It tells

NOTHING about the value (sum) of the series. The table below details this test.

The Integral Test

If a_n ≥ 0 for all n, a_k = f(k) where f is a continuous, decreasing function,

Then ∞

∑ a_n converges if and only if ∫ f(x) dx converges.

1

Restrictions:

Note: if ∑ a_n must be a positive term series.

Note: a_n+1 < a_{n ,} f(k+1) < f(k)

Click here for a list of theorems (tests) to determine convergence or divergence

of infinite series.

Click here for examples involving positive term series.

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