Comparison Tests

A series with positive terms has increasing partial sums. They converge exactly when they stay below some fixed number. So a series of positive terms can be judged against another series whose behavior is already known.

Direct comparison

The partial sums of the smaller series stay below those of the larger one. If the larger series has a finite sum, the smaller one is trapped under it. If the smaller one grows without bound, it pushes the larger one up with it.

The other two cases prove nothing. A series smaller than a divergent one may converge or diverge, and so may a series larger than a convergent one. The usual comparison series are geometric series and p-series.

Limit comparison

Sometimes a series behaves like a simpler one, but the inequality points the useless way. The limit comparison test compares the two series through the ratio of their terms.

When the ratio approaches a positive number

L

, it's eventually between

\frac{L}{2}

and

2L

. So from some point on,

a_n

is trapped between two constant multiples of

b_n

, and direct comparison works in both directions. To choose

b_n

, keep the fastest-growing part of the numerator and of the denominator.

Example 3

Decide whether

\sum_{n=1}^{\infty} \frac{1}{n^2 - 5n + 7}

converges.

The discriminant of

n^2 - 5n + 7

25 - 28 < 0

, so the denominator is always positive and every term is positive. For large

n

the denominator behaves like

n^2

, so compare with

b_n = \frac{1}{n^2}

Direct comparison with

\frac{1}{n^2}

fails here. For

n \ge 2

n^2 - 5n + 7 < n^2

, so

a_n > \frac{1}{n^2}

, and being larger than a convergent series proves nothing. Take the limit of the ratio instead.

\begin{aligned} &\lim_{n\to\infty} \frac{1/(n^2 - 5n + 7)}{1/n^2} \\[4pt] &\qquad = \lim_{n\to\infty} \frac{n^2}{n^2 - 5n + 7} \\[4pt] &\qquad = \lim_{n\to\infty} \frac{1}{1 - 5/n + 7/n^2} = 1 \end{aligned}

Since

0 < 1 < \infty

and

\sum \frac{1}{n^2}

converges (a p-series with

p = 2 > 1

), the series converges by limit comparison.

The ratio of the terms for n = 1 to 40. It's above 1 from n = 2 on, and it approaches 1.