Hi I need problems [ 1-5, ] solved.

Please write full steps in order to understand. Thank you

1 1. Consider the sequence {an} _1 and series 2% where an 1 2: 1121 (a) Find the rst four terms of the sequence can. (b) Find the rst four partial sums of the series 2 an. (o) Does the sequence converge? (d) It is not easy to nd a general formula for 3,, but based on your answers to (b) or other ideas you know about series, does the series converge? 2. Determine whether the following series converge absolutely, converge conditionally, or diverge: 3??? +1 (C) Z;2n3+'n. 11:1 3. Find the center, IOC, and R of the following power series: 0 2n+1($ _ 5):; (a) \":1 n 0 (m+1)\" (b) \"2:; (2n)! 4. (a) Find a power series representation for f(a:) = 31n(7 23:) given that 31n(7 23:) is an 7 23: (b) Use the power series representation for arctan(:c ) to nd a power series representation for antiderivative of . What is R for this p.s. representation? 9(17): 411:3 arctan ( 3). What is R for this p. s. representation? 5. Find the sums of the following series. (Hint: You'll want to use your answers to # 4 above for some of them.) (a) 2301"); 8"\" n=0 6(2\") 13):: 0(n + 1)7\"+1 (2)2n+4 (C) Z:(_1)n3,2n+(1(2n+1) n=0