# Answer to Question #340457 in Real Analysis for Yaa

Question #340457

) Give an example to show that if the convergence of 􏰄 an is conditional and (bn) is a bounded

sequence, then 􏰄 anbn may diverge.

1
2022-05-16T12:25:36-0400

ANSWER :"\\sum_{n=1}^{\\infty}a_{n}=\\sum_ {n=1}^{\\infty}\\frac{(-1)^{n } }{n},\\, \\, b_{n}=(-1)^{n}"

EXPLANATION The series "\\sum_ {n=1}^{\\infty}\\frac{(-1)^{n } }{n}" is an alternating series, since the sequence "a_{n}=(-1)^{n}c_{n}" and "c_{n}= \\frac {1}{n\n\n}" is decreasing , "\\lim _{n\\rightarrow\\infty}c_{n}=0" . The series "\\sum_ {n=1}^{\\infty}\\frac{1 }{n}" diverges, because it is a "p-" series for "p=1" , hence the series "\\sum_ {n=1}^{\\infty}\\frac{(-1)^{n } }{n}" converges conditionally . The sequence "b_{n}=(-1)^{n}" is a bounded sequence , since "-1\\leq b_{n}\\leq 1" for all "n\\in\\N" ("b_{2n }=1,\\, b_{2n-1}=-1" ). Since "a_{n}\\cdot b_{n}=\\frac {1}{n}" , then the series "\\sum_{n=1}^{\\infty}a_{n}\\cdot b_{n}=\\sum_ {n=1}^{\\infty}\\frac{1 }{n}" diverges.

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