Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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4.3 CONVERGENCE OF INFINITE SERIES


4.3.2 Convergence of a series containing only real positive terms

As discussed above, in order to test for the absolute convergence of a series

un, we first construct the corresponding series



|un|that consists only of real

positive terms. Therefore in this subsection we will restrict our attention to series


of this type.


We discuss below some tests that may be used to investigate the convergence of

such a series. Before doing so, however, we note the followingcrucial consideration.


In all the tests for, or discussions of, the convergence of a series, it is not what


happens in the first ten, or the first thousand, or the first million terms (or any


other finite number of terms) that matters, but what happensultimately.


Preliminary test

A necessarybut not sufficientcondition for a series of real positive terms



un

to be convergent is that the termuntends to zero asntends to infinity, i.e. we


require


lim
n→∞

un=0.

If this condition is not satisfied then the series must diverge. Even if it is satisfied,


however, the series may still diverge, and further testing is required.


Comparison test

The comparison test is the most basic test for convergence. Let us consider two


series



unand


vnand suppose that weknowthe latter to be convergent (by

some earlier analysis, for example). Then, if each termunin the first series is less


than or equal to the corresponding termvnin the second series, for allngreater


than some fixed numberNthat will vary from series to series, then the original


series



unis also convergent. In other words, if


vnis convergent and

un≤vn forn>N,

then



unconverges.
However, if


vndiverges andun≥vnfor allngreater than some fixed number

then



undiverges.

Determine whether the following series converges:
∑∞

n=1

1


n!+1

=


1


2


+


1


3


+


1


7


+


1


25


+···. (4.7)


Let us compare this series with the series


∑∞

n=0

1


n!

=


1


0!


+


1


1!


+


1


2!


+


1


3!


+···=2+


1


2!


+


1


3!


+···, (4.8)

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