Let (an)n>1 be the sequence of real numbers defined by a1 = 5, and for n 2 1 an+1 = an - an + 1 Prove that the series an converges and find its sum. n=1 Hint: Please consider the following steps . Let bn = 1, then . Show that bn+1 = b2 - bn +1 . Show that on (bn+1 - bn) = (bn - 1) (bn+1 - 1) . Show that 1 1 an = bn - 1 bn+1 - 1 . Notice that b1 = 2 and then show that Sn = ak but 1 - . Prove that on - co as n - co using the steps below i) Use induction on n to show that bn > n Hint: Notice that for n > 2, if bn > n, then be+1 = bn (bn -1) +1>n+1. ii) Use induction on n to show that (bn),>1 is increasing Hint: Notice that bn+1 - bn = by + 1 . Compute an lim Sn n-+ 0o n=1