104 convergent series

=

1

|

The proof of this test relies on making clever compari-

son to a geometric series. (If, for all terms,

n

√

–

an≤rfor

some value r, then an≤rnand a comparison can be

made.) This test is often used if the series contains

terms involving exponents. For example, consider the

series . Here the nth term of the series is

given by , and we have:

. By the root test, the series must

converge.

The Integral Test

This test applies only to series with positive terms.

Suppose the terms of a series are given

by a formula an= f(n),where the function f(x) is

continuous, positive, and decreasing for x≥1:

i. If the

IMPROPER INTEGRAL

∫∞

1f(x)dx con-

verges, then the series converges.

ii. If the improper integral ∫∞

1f(x)dx diverges,

then the series diverges.

This can be proved geometrically by drawing rectan-

gles of width 1 just above and just below the graph of

y= f(x) for x ≥1, and then comparing the total area

of all the rectangles with the area under the curve.

To illustrate the test, consider the series . Since

converges, we have that

the series converges. In general, one can establish

in this way the p-series test.

The p-Series Test

A series of the form with pa real

number converges if p > 1and diverges if p ≤1.

A series of the form is called a p-series.

Absolute Convergence Test

Suppose is a series with both positive

and negative terms. If the corresponding series

with all terms made positive converges,

then the original series also converges.

One can use any of the first six tests described above

to determine whether or not converges. The

validity of the absolute convergence test is established

in the discussion on

ABSOLUTE CONVERGENCE

.

Since we have established, for example, that

converges, the absolute

convergence test now assures us that the variant series

also

converges (as does any other variation that involves the

insertion of negative signs).

The absolute-convergence test does not cover all

cases. It is still possible that a series with negative

terms, , might converge even though

diverges. This phenomenon is called

CONDITIONAL

CONVERGENCE

.

Alternating-Series Test

If the terms of a series alternate in sign

and satisfy

i. a1 ≥ a2 ≥ a3 ≥ …

ii. an→0

then the series converges.

(See

ALTERNATING SERIES

.) This test shows, for

example, that the alternating

HARMONIC SERIES

converges even though the