Answer Happy

1. Determine whether the series n=1∑∞​(−1)n+1n3+5n​ converges absolutely, converges conditionally, or diverges. a. converges absolutely b. converges conditionally c. diverges d. cannot be determined 2. The function (1−x)21​ has power series representation a. n=0∑∞​x2n for ∣x∣<1. b. n=1∑∞​nxn−1 for ∣x∣<1. c. n=0∑∞​n+1xn+1​ for x∈[−1,1). d. n=0∑∞​(−1)n(n+1xn+1​) for x∈(−1,1]. e. None of the above 3. The function f(x)=−ln(1−x) has power series representation a. n=1∑∞​nxn−1 for ∣x∣<1. b. n=0∑∞​(−1)nnxn−1 for ∣x∣<1. c. n=0∑∞​(−1)n(n+1xn+1​) for x∈(−1,1]. d. n=0∑∞​n+1xn+1​ for x∈[−1,1). e. None of the above