“Notice that if , then and , whereas if
, then and .
(a) If is absolutely convergent, show that both of the
series and are convergent.
(b) If is conditionally convergent, show that both of the
series and are divergent.
44. Prove that if is a conditionally convergent series and
is any real number, then there is a rearrangement of
whose sum is . [Hints: Use the notation of Exercise 43.
an an

0 a
n
an
0
an an
0 an
0
an

an

an

a
n


an

an

a
n


an
r

an
r
Take just enough positive terms so that their sum is greater
than . Then add just enough negative terms so that the
cumulative sum is less than . Continue in this manner and use
Theorem 11.2.6.]
45. Suppose the series is conditionally convergent.
(a) Prove that the series is divergent.
(b) Conditional convergence of is not enough to determine whether is convergent. Show this by giving an
example of a conditionally convergent series such that
converges and an example where diverges.
r an

r

an

n
2
an

an

nan

nan

nan
an

We now have several ways of testing a series for convergence”