8.3. SERIES OF REAL NUMBERS
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Activity 8.12 illustrates how we can compare one series with positive terms to another
whose behavior (that is, whether the series converges or diverges) we know. More generally,
suppose we have two series
a k and
b k with positive terms and we know the behavior
of the series
a k . Recall that the convergence or divergence of a series depends only on
what happens to the terms of the series for large values of k, so if we know that a k and b k
are essentially proportional to each other for large k, then the two series
a k and
b k
should behave the same way. In other words, if there is a positive finite constant c such
that
lim
k→∞
b k
a k
= c,
then b k ≈ ca k for large values of k. So
b k ≈
ca k = c
a k .
Since multiplying by a nonzero constant does not affect the convergence or divergence of
a series, it follows that the series
a k and
b k either both converge or both diverge. The
formal statement of this fact is called the Limit Comparison Test.
The Limit Comparison Test. Let
a k and
b k be series with positive terms. If
lim
k→∞
b k
a k
= c
for some positive (finite) constant c, then
a k and
b k either both converge or both
diverge.
In essence, the Limit Comparison Test shows that if we have a series
p(k)
q(k) of rational
functions where p(k) is a polynomial of degree m and q(k) a polynomial of degree l, then
the series
p(k)
q(k) will behave like the series
k m
k l . So this test allows us to quickly and
easily determine the convergence or divergence of series whose summands are rational
functions.
Activity 8.13.
Use the Limit Comparison Test to determine the convergence or divergence of the
series
3k 2 + 1
5k 4 + 2k + 2
.
by comparing it to the series
1
k 2 .
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