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