(a)Carefullydeterminetheconvergenceoftheseriesn. The series isSolution:SOLUTIONNote that for both of these series the terms alternate in sign,∞∑n=1(-1)n5•A. absolutely convergent•B. conditionally convergent•C. divergent(b)Carefullydeterminetheconvergenceoftheseries∞∑n=1(-1)n5n. The series is•A. absolutely convergent•B. conditionally convergent•C. divergentand the magnitudes of successive terms decrease, so that we areable to apply the alternating series test.(a)For∞∑n=1(-1)n5n, we have limn→∞15n=0. Thus, by the alternat-ing series test, we know that this series converges (it is at leastconditionally convergent). Because∞∑n=115ndiverges (because itis a constant times ap-series withp=1), is only conditionally,not absolutely, convergent.(b)For∞∑n=1(-1)n5n, we have limn→∞15n=0. Thus, by the alter-nating series test, we know that this series converges (it is atleast conditionally convergent). Because∞∑n=115nalso converges(by comparison to ap-series or by the integral test), our originalseries must be not only conditionally convergent but absolutelyconvergent.Correct Answers:•B•A
10.(1 point) For each of the series below select the letterfrom a to c that best applies and the letter from d to k that bestapplies. A possible answer is af, for example.