4.3 CONVERGENCE OF INFINITE SERIES
4.3.2 Convergence of a series containing only real positive termsAs 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 realpositive 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 ofsuch 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 testA necessarybut not sufficientcondition for a series of real positive terms
∑
unto 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 testThe comparison test is the most basic test for convergence. Let us consider two
series
∑
unand∑
vnand suppose that weknowthe latter to be convergent (bysome 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 andun≤vn forn>N,then
∑
unconverges.
However, if∑
vndiverges andun≥vnfor allngreater than some fixed numberthen
∑
undiverges.Determine whether the following series converges:
∑∞n=11
n!+1=
1
2
+
1
3
+
1
7
+
1
25
+···. (4.7)
Let us compare this series with the series
∑∞n=01
n!