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Analysis Homework #8 due Thursday, Mar. 31 1. Test each of the following series for convergence: ( ) ∞ ∑ 1 1 · log 1 + , n n n=1 ∞ ∑ (−1)n e1/n n=1 n . 2. Find the radius of convergence for each of the following power series: ∞ ∑ nxn n=0 3n , ∞ ∑ (n!)2 n ·x . (2n)! n=0 3. Assuming that |x| < 1, use the formula for a geometric series to show that ∞ ∑ n=0 nxn = x . (1 − x)2 4. Let α ∈ R be fixed. Find the radius of convergence for the binomial series f (x) = ∞ ∑ α(α − 1)(α − 2) · · · (α − n + 1) n=0 n! · xn . • You are going to work on these problems during your Thursday tutorial (at 4 pm). • When writing up solutions, write legibly and coherently. Use words, not just symbols. • Write your name and then MATHS/TP/TSM on the first page of your homework. • Your solutions may use any of the results stated in class (but nothing else). • NO LATE HOMEWORK WILL BE ACCEPTED.