1. Consider the series 2n+3 2n+5 Decide en en+1 n=1 [2 marks] whether it converges or diverges; if it converges, find the limit. 2.

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1. Consider the series 2n+3 2n+5 Decide en en+1 n=1 [2 marks] whether it converges or diverges; if it converges, find the limit. 2. f(2) = 82-3n 27" n=1 Throughout it, let be the function defined by [3 marks] (a) What is the interval of convergence of the series defining f? You do not need to check the endpoints of the interval you may assume it is an open one. (b) By recognising the power series defining f as a geometric series and taking the limit of that series, it is possible to give a formula which has the same value as f at every point in the interval of convergence, but which does not mention any series. Find such a formula. (c) First, find a power series which is equal to f'(x) on the interval of convergence you found in (a). Then, give a formula which is equal to f'(x) on that interval, but which does not mention any series.