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Guidelines for evaluating convergence or divergence of Infinite Series – Test Options

In a Nut Shell: You have seven tests available to evaluate the convergence or

divergence of infinite series.

The integral test, the comparison test, and the limit comparison test apply to positive

term series only.

The alternating series test applies only to alternating series.

The nth term test, the ratio test, and the root test apply to both positive term and alternating

term series.

Strategy: Start by classifying the type of series. Is it a positive term or an alternating

term series? Is it similar to a harmonic series, a p-series, or a geometric series? Use

this information to proceed with tests appropriate to these series.

The quickest test is the nth term test. If you can show that lim a_n ≠ 0, then you know

n→ ∞

that the series diverges. Typically this does not happen but it is worth a try.

If the series is a positive term series check to see if it is of the form of a harmonic

series, a p-series, or a geometric series. Normally this involves manipulation of the

given series, Σ a_n , to be tested. You can then use the comparison tests.

If the series is an alternating series of the form Σ(-1)ⁿ a_n or Σ(-1)ⁿ⁺¹ a_n, then

try the alternating test, the ratio test, or the root test.

If the series involves factorials, products, or powers, then try the ratio test. Note,

however, that the ratio test fails to provide information for series involving rational

or algebraic functions since | a_n+1/ a_n | → 1 as n → ∞.

If Σ a_n is of the form Σ( a_n ) ⁿ, then try the root test.

∞

If a_n = f(n) where ∫ f(x) dx is easily evaluated, then the integral test may be useful

provided the hypotheses for this test are satisfied.

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc