Derivatives and Anti-Derivatives of Taylor Series
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Problems 3
- (A-Star Math Tournament 2015, Calculus Tiebreaker #3) Compute: ∑k=0∞(k+1)3k!.\sum_{k=0}^{\infty}\frac{(k+1)^3}{k!}.k=0∑∞k!(k+1)3. Round to the nearest hundredth.
- Let S=∑n=1∞1n⋅2n−1S = \sum_{n=1}^{\infty} \frac{1}{n\cdot 2^{n-1}}S=∑n=1∞n⋅2n−11. Find eSe^SeS. Source: HMMT February 1998
- Compute 11+22+34+48+⋯+n2n−1+⋯\frac{1}{1}+\frac{2}{2}+\frac{3}{4}+\frac{4}{8}+\cdots+\frac{n}{2^{n-1}}+ \cdots11+22+43+84+⋯+2n−1n+⋯.