# 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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