Notice that if , then and , whereas if, then and .{a} If is absolutely...

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