# A calculus problem by Rocco Dalto

Calculus Level pending

Let $${\bf K }$$ be a positive integer and $${\bf X > 1}$$

$$If$$ $${\bf \sum_{N = 1}^{\infty} \frac{N^K}{X^N} }$$ $$=$$ $${\bf \frac{\sum _{J = 0}^{K - 1} b_{K - J} * X^{K - J}}{(X - 1)^{K + 1} } }$$

where $${\bf b_{1} = b_{K} = 1}$$ and each constant $$\bf b_{i}$$ where $${\bf (2 <= i <= K - 1) }$$ is a positive integer

$$then$$

$${\bf \sum_{N = 1}^{\infty} \frac{N^{K + 1}}{X^N} }$$ $$=$$ $${\bf \frac{X^{K + 1} + (\sum_{J = 0}^{K - 2} ((K - J) * b_{K - J} + (J + 2) * b_{K - J - 1}) * X^{K - J}) + X}{(X - 1)^{K + 2}} }$$

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