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MATH 101 V2A March 20th – Practice problems Hints and Solutions Practice Problems: 1. Find a power series which converges on the interval [2, 8). Solution: The series n ∞ X 1 x−5 n 3 n=1 is one example. 2. Find a series which converges to e. You may assume that there is a power series ∞ X an xn which n=0 converges to ex for all x < R (for some R > 1), but your answer cannot contain the coefficients an (i.e. you must find an expression for an in terms of n). Note: This problem was given before we covered Taylor series, so the solution does not use any Taylor series. Solution: Since d x dx e = ex , we get that ∞ X an nxn−1 = n=1 X an xn , n=0 and so we can conclude that (n + 1)an+1 = an for all n ≥ 0. We also know that Z x et dt = ex − 1, 0 so ∞ X n=0 an ∞ X xn+1 = −1 + an xn . n+1 n=0 In particular, this means that a0 − 1 = 0. So, a1 = 1 in general, an = n! . 1 1 · a0 = 1, a2 = 1 2 · a1 = 1 2! , a3 = 3. Determine the radius and interval of convergence for each of the following series. (a) ∞ X (−1)n . n5 5 n n=1 (b) ∞ X (1 + 3n )xn . n! n=0 Solution: See the solutions to the practice problems from March 18th. 1 3 · a2 = 1 3! and,