\begin{matrix}c \\ ◯ \end{matrix} Stephen A. Fulling, Texas A&M Univ., 1997

Testing a Series ∑^{∞} a_{n} for Convergence or Divergence

I. Do the terms tend to zero? [That is, a_{n} → 0 as n → ∞.] If not, the series

diverges. II. Is the series recognizable as one for which the answer is known? Examples:

(a) ∑_{n=1}^{∞} \frac{2}{3^{n}} = \frac{2}{3} ∑_{n=0}^{∞} \frac{1}{3^{n}} converges to \frac{2}{3} \frac{1}{1 − \frac{1}{3}} = 1 (geometric series with |r| < 1).

(b) ∑_{n=0}^{∞} \frac{10^{n}}{n!} x^{2n} converges to e^{10x^{2}} (Maclaurin (Taylor) series of e^{z} with z = 10x^{2}).

(c) ∑_{n=1}^{∞} \frac{3}{n^{2}} converges (p-series with p > 1).

(d) ∑_{n=0}^{∞} \frac{1}{n + 2} = ∑_{m=2}^{∞} \frac{1}{m} [let m = n + 2] diverges (harmonic series).

III. Are all the terms positive? If so, go to IV. If not, try the following strategies: A. Are all the hypotheses of the alternating series theorem satisfied?

a_{n} = (−1)^{n}|a_{n}|; |a_{n}| is decreasing ; |a_{n}| → 0 .

If so, the series converges (and the error is bounded by the magnitude of the first term omitted).

B. Try to show that the series is absolutely convergent [that is, ∑ |a_{n}| converges]

by one of the methods under IV. IV. If a_{n} ≥ 0 (at least for all sufficiently large n), try the following strategies:

A. If the terms contain factorials, or n in an exponent, try the ratio or root test. (In particular, these methods are used to find the radius of convergence of a power series.)

1. If a_{n} contains factorials, the ratio test works best. Example: a_{n} = \frac{3^{n}n^{2}}{n!} .

2. If n appears in both an exponent and its base, the root test works best.

Example: a_{n} = \frac{2^{n}}{n^{n}} .

B. If a_{n} is a rational function [for example, a_{n} = \frac{n^{2}+3}{n^{3}−4} ], the ratio and root tests won’t work, because the limit involved turns out to be 1. Try the following strategies:

1. Use the comparison test or limit comparison test to replace the series by one for which you know the answer (see II) or can find the answer by the integral test (see below). In choosing a comparison series ∑ b_{n}, keep (at least as a first step) from each factor in a_{n} the term that grows fastest as n → ∞. Examples:

(a) If a_{n} = \frac{1}{n^{2}+1} , take b_{n} = \frac{1}{n^{2}} . Since a_{n} < b_{n} and ∑ b_{n} is a convergent p-series, ∑ a_{n} converges by the comparison test. Alternatively, since a_{n}/b_{n} → 1 (a finite, nonzero limit) as n → ∞, and ∑ b_{n} converges, the limit comparison test shows that ∑ a_{n} converges.

(b) If a_{n} = \frac{\sqrt{n}}{n− \frac{1}{2}} , take (as a first step) b_{n} = \frac{\sqrt{n}}{n} . We have a_{n} > b_{n}.

Furthermore, b_{n} ≥ c_{n}, where c_{n} = \frac{1}{n} , and the harmonic series ∑ c_{n} diverges. Therefore, ∑ a_{n} diverges, by the comparison test. Alternatively, try the limit comparison test: a_{n}/c_{n} = \sqrt{n} \frac{n}{n− \frac{1}{2}} approaches infinity, and the harmonic series diverges, so ∑ a_{n} diverges.

(c) The signs of the constant terms in the two previous examples were chosen to make the comparison test easy to apply. For a_{n} = \frac{1}{n^{2}−1}

or a_{n} = \frac{\sqrt{n}}{n+ \frac{1}{2}} , a suitable comparison series, with the inequalities running in the right direction, is not so obvious; therefore, the limit comparison test is more convenient in those cases.

(d) If a_{n} = (2 + cos n)/n^{2}, neither term in the numerator dominates as n → ∞, so the limit comparison test is hard to use. But a_{n} ≤ b_{n} = 3/n^{2}, so the comparison test shows that ∑ a_{n} converges.

2. Use the integral test if a_{n} = f(n), where f(x) is positive and decreasing and ∫_{1}^{∞} f(x) dx can be evaluated. Example:

∫_{1}^{a} \frac{ln x}{x^{p}} dx

can be evaluated by the substitution u = ln x. (This integrand isn’t decreasing at first, but it is for sufficiently large x, which is all that matters.) The integral converges if and only if p > 1, so

∑_{n=1}^{∞} \frac{ln n}{n^{p}}

converges or diverges just like a p-series without the logarithmic factor. (This conclusion can also be reached by comparison or limit comparison arguments.)