University Mathematics Handbook (2015)

VI. Series

Chapter 2. Positive Series

A positive series is a series of all-positive terms, except, possibly, a finite number of terms.

2.1  Comparison Tests

Given two positive series .

a.  First Comparison Test: if, for all starting from a certain number , then:

1.  From the convergence of series (B), there follows the convergence of series (A).

2.  From the divergence of series (A), there follows the divergence of series (B).

b.  Second Comparison Test: Let the limit exists:

1.  If , then series (A) and (B) are either both convergent or both divergent.

2  If , then from the convergence of series (B), there follows the convergence of series (A).

3.  If , then from the convergence of series (A), there follows the convergence of series (B).

c.  Third Comparison Test: if, for all starting from a certain number

,

then:

1.  From the convergence of series (B), there follows the convergence of series (A).

2.  From the convergence of series (A), there follows the convergence of series (B).

2.2  D'Alembert's Ratio Test

a.  If, starting from a certain n, for the terms of series, , , there holds:

1.  , then the series is convergent.

2.  , then the series is divergent.

b.  The Limit Form of Ratio Test: if the limit exists, then, when:

1.  , the series is convergent.

2.  , the series is divergent.

3.  , the test is inconclusive regarding the convergence or divergence of the series.

2.3  Cauchy's Root Test

a.  If, starting from a certain n, for the terms of series, , , there holds:

1.  , then the series is convergent,

2.  , then the series is divergent.

b.  The Limit Form of Root Test: if the limit exists, then, when:

1.  , the series is convergent.

2.  , the series is divergent.

3.  , the test is inconclusive regarding the convergence or divergence of the series.

2.4  Integral Test

a.  If is a positive function, non-increasing in domain , and equals the value of function at , then the series and integral are either both convergent or both divergent (see IV, 3.1).

b.  If a series is convergent, then, for its remainder , there holds