University Mathematics Handbook (2015)

VI. Series

Chapter 3. General Series

Series containing an infinite number of positive terms and an infinite number of negative terms are called general series.

3.1  Absolute and Conditional Convergence

a.  Series is absolutely convergent if series is convergent.

b.  An absolutely convergent series is convergent.

c.  A convergent, but not absolutely convergent series, is conditionally convergent.

d.  Examples:

1.  The series is absolutely convergent, since series is convergent.

2.  Leibniz series is conditionally convergent since the series itself is convergent and series is divergent.

3.2  Operations on the Terms of a Series

a.  If series converges, then, all series resulting from grouping its terms in sets, without changing the order of terms from the original series, converges to the same sum.

The inverse is incorrect. That is, if a series with groups is convergent, the series without groups is not necessarily convergent.

Example: The series is convergent, yet the ungrouped series is divergent.

b.  Cauchy's Theorem

If series is absolutely convergent, then, all series resulting from any changer of the order of its terms absolutely converges to the same sum.

c.  Riemann Theorem

If series is conditionally convergent, then its terms can be arranged in such a way that the rearranged series will converge to a predetermined sum , or even diverge.

Example: Let's rearrange the terms of a Leibniz series

the following way:

We denote by and the partial sums of the Leibniz series and the rearrange series, respectively:

Therefore, the rearrange series converges to a different sum, .

3.3  General Series Convergence Tests

a.  Leibniz Test

If sequence is monotonic, positive, and , then, the sign-changing series is:

1.  Convergent.

2.  For its sum , there holds: .

3.  Its remainder, .

b.  Dirichlet Test

If any term of the series can be presented as the product of two terms, , such that:

1.  Sequence is monotonic and converges to zero.

2.  All partial sums of series are common bounded, that is, there exists number such that for all , there holds , then series is convergent.

c.  Abel's Test

If any term of the series can be presented as the product of two terms, , such that:

1.  Sequence is monotonic and bounded.

2.  Series is even conditionally convergent, then series is convergent.

3.4  Product of Series

a.  A series consisting of all products when is called the product of series , .

b.  If both series are absolutely convergent, then their product absolutely converges to and: