Suppose we have a function f defined by the power series f(x) = > ak(x - a)k k=0 on its interval of convergence I. It turns out that f is differentiable on I, and f' (z) = kak(x - a)k-1, for a E I. =1 That is, differentiating a function defined by a power series is the same as differentiating the power series term-by-term inside the interval of convergence. We can use this to obtain closed-form expressions for some new power series from old ones we already know.Likewise, if we again have f defined by the power series f() = ak(x - a)k k=0 on the interval of convergence I, then f is integrable on ], and f(x) dr = ak (x - a) + + C, for zel, k +1 k=0 and Cis a constant of integration. That is, as with differentiation, integrating a function defined by a power series is the same as integrating the power series term-by-term inside the interval of convergence.In the following, use the fact that we know and some clever substitutions to obtain closed-form expressions for the following related infinite series: (i) 1-at-8+=)(1= (ii) 1+z te to=) 00 (iii) 1-x +x -1 (-1)1 2k = Now, if we integrate the last formula above (noticing that there is no constant of integration on either side), we get: (-1)*24+1 3 2k + 1 This series was originally called Gregory's series, named after the Scottish mathematician James Gregory (1638- 1675). Since it was first discovered by the Indian mathematician Madhava of Sangamagrama (c.1340 - c. 1425), it is also referred to as the Madhava-Gregory series. When we substitute = = 1 into the Madhava-Gregory series, we get the famous and surprising formula 09 (-1)k 2k + 1 which has many names, one of which is the Madhava-Leibniz formula for #, named for German mathematician Gottfried Leibniz (1646-1716)