James Stewart Quote
Notice that if , then and , whereas if, then and .(a) If is absolutely convergent, show that both of theseries and are convergent.(b) If is conditionally convergent, show that both of theseries 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 an an 0an an 0 an 0 ananana nanana nanranrTake just enough positive terms so that their sum is greaterthan . Then add just enough negative terms so that thecumulative sum is less than . Continue in this manner and useTheorem 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 anexample of a conditionally convergent series such thatconverges and an example where diverges.r anrann2anannannannananWe now have several ways of testing a series for convergence
Notice that if , then and , whereas if, then and .(a) If is absolutely convergent, show that both of theseries and are convergent.(b) If is conditionally convergent, show that both of theseries 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 an an 0an an 0 an 0 ananana nanana nanranrTake just enough positive terms so that their sum is greaterthan . Then add just enough negative terms so that thecumulative sum is less than . Continue in this manner and useTheorem 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 anexample of a conditionally convergent series such thatconverges and an example where diverges.r anrann2anannannannananWe now have several ways of testing a series for convergence