Hi I need problems [ 1-5, 7 ] solved.

Please write full steps in order to understand. Thank you

Series Review Problems 1. For the following problem, think about the difference between a sequence, a series, and the sequence of partial sums. 2n 7 (a) Suppose you have a sequence with terms an = 4n 5. Does this sequence converge or diverge? If it converges, what does it converge to? 00 _ 2n 7 . . . (b) Does the series 2 4 5 converge or diverge? If it converges, what does it converge to? n __ 01:1 00 . . . 2n 7 . (c) Suppose you have a series 2b,, whose nth partial sum is Sn 2 4n 5 Does the series 00 n=1 2 bn converge or diverge? If it converges, what does it converge to? 71:1 2. Give examples of series, one for each of the following conditions. Explain why you know the series you came up with satises that condition. (a) A series that converges absolutely. (b) A series that converges conditionally. (c) A series that diverges. 4 3. Consider the series: 2 \"=1 \\/71 + 1 (a) Try to use the comparison test on this series and explain why it does not work. (b) Explain why you can use the integral test and then use it to determine whether the series converges or diverges. 4. (a) Consider :01 + 2)!%. Find the center, IOC, and R. 71.21 (b) Suppose X Cn(IL' 2)" is a power series with 100 = (1, 5]. Does Z cn(1)\" converge or n=1 71:1 00 diverge? Does 2 CH4\" converge or diverge? n=1 . . . . 5 . 5. (a) Find a power series representation of the function f (1:) = 3 + 4%. What is R? 20 (b) Use your answer to nd a p.s. representation of 9(1)) 2 m. (Hint: g(1:) = f'(m).) ac 00 5 4 3 n (c) Find the sum of the series: 2 (2711) "=1 $1.4, (\"hr Sales 1 :L') (3' m - ) fan 5A1 Cow ~+ tr WW) 9:3" can\" 61* 7