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VJIMC 2015 Category II, Problem 3

Source: VJIMC2015

August 10, 2015
SummationConvergenceinfinite sumcollege contestsreal analysis

Problem Statement

Problem 3 Determine the set of real values of xx for which the following series converges, and find its sum: ∑n=1∞(∑k1,k2,…,kn≥01⋅k1+2⋅k2+…+n⋅kn=n(k1+…+kn)!k1!⋅…⋅kn!xk1+…+kn) .\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ .
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