12 1 1 Consider the Z % (a) Use any test for convergence/divergence to show that the series converges. oo _ . , \\f 12 1 n , , _ (b) It Is pOSSIble to show that the sum of the serIes E l ) Is 1r, In other words, the serIes 3'\"(2n + 1) =0 converges to the number 1r. (You do NOT need to prove this. but it can be done somewhat easily using a Taylor series expansion of arctan :3.) Suppose you want to use a partial sum of this series to estimate the value of at 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.) Hint: Use Theorem 5.14. \\/_(4n)l(1103 + 26390n) 9801- 3964\"(n!)4 .This series was discovered by the extraordinary Indian 1 mathematician Srinivasa Ramanujan (1887-1920). This series converges to .(You do NOT need to prove that. and it is much more difficult than finding the sum of the series in problem 1. ) This series has been used to compute or to over 17 million digits (which was a world record at the time). 2. Now consider the series 2 (1103 + 26390n) (a) -Use any test for convergence/divergence to show that the series 2 flggolll. 3964\"(n!)4 converges. k l 11 2 (b) -The parts 5...... I... this s. = Z \"(333,? ,Sim'ff'fm'). 71:0 ' Use a calculator to evaluate SL0 and Sil. and write down as many digits as your calculator can display. How many digits are the same as the digits of it? Note: 1r :3 31415926535 8979323846 2643383279... THEOREM 5.14 Remainders in Alternating Series Consider an alternating series of the form 0O (-1)nbnor (-1)"bn n=1 n=1 that satisfies the hypotheses of the alternating series test. Let S denote the sum of the series and SN denote the Nth partial sum. For any integer N 2 1, the remainder RN = S - SN satisfies | RN| S bN+1