Q8. (a) Define the radius of convergence of a complex power series an. (b) Compute the radius of convergence of the power series 3/7/20 2n + 3 ∞ Σ(-1)", n=1 ∞ 2n+1 2 n=0

8. Determine whether the series is divergent or convergent, and if convergent, then absolutely or conditionally : (b) (a) n=1 (-1)" n√1+n 1+ n² Σ(-1)" en 22n+3 (0) Σ sin(n) n=1 710 37. (2 + 1)²

Show all of your work and mention what formula/theorem you are using. (a) We are going to find a formula for the sum of the series i(i +1) i=1 1 1 i(i + 1) i+1" First observe that - (i) Find a closed form formula for the nth partial sum s, in terms of n (but without any sigma). (ii) Find lim sp. (b) Consider the series (i) Use properties of logarithm to find a closed form formula for the nth partial sum in terms of n (but without any sigma). (ii) Does the series converge or diverge?

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