Problems: . . miun . x/H)\" 1. Con5|der the series \"EMT Note. this was changed from \"2:31 37'9\" +1). (3) (3 marks) Use any test for convergence/divergence to show that the series converges. x/l)\" \"=0 3"(2n + 1) converges to the number 11'. (You do NOT need to prove this. but it can be done somewhat easily using a Taylor series expansion of arctan 3.) DO (b) (2 marks) It is possible to show that the sum of the series 2 is 11'. in other words, the series Suppose you want to use a partial sum of this series to estimate the value of 7r to an accuracy of within 0.0001. Would using the first 8 terms of the series be enough to ensure you get an accuracy of within 0.0001? (8 terms means the terms where n = 0, 1,2, 3, ...., 7.) Note: This was changed from "7 terms". Hint: Use Theorem 5.14. . This series was discovered by the extraordinary Indian 0 1 2. Now consider the series Z: W :0 9801 - 3964\"(n!)4 1 mathematician Srinivasa Ramanujan (1887-1920). This series converges to -. (You do NOT need to prove that, 71' and it is much more difficult than finding the sum of the series in problem 1.) This series has been used to compute it to over 17 million digits (which was a world record at the time). Jean) !(1103 + 26390n) on (a) (3 marks) Use any test for convergence/divergence to show that the series 2 9801 3964\"( [)4 - n. \"=0 converges. '9 I (b) (2 marks) The partial sums for this series are Sr; = 2 WW. n=0 Use a calculator to evaluate SL0 and 317 and write down as many digits as your calculator can display. How many digits are the same as the digits of 71'? Note: 11' % 31415926535 8979323846 2643383279... Bonus: Watch the movie \"The man who knew infinity" (2015) about the life of Srinivasa Ramanujan