Poly and logarithms

Calculus Level 4

∑n=1∞13n,∑n=1∞n3n,∑n=1∞n23n,∑n=1∞n33n \sum_{n=1}^\infty \dfrac1{3^n} , \quad \sum_{n=1}^\infty \dfrac n{3^n} , \quad \sum_{n=1}^\infty \dfrac{n^2} {3^n} , \quad \sum_{n=1}^\infty \dfrac{n^3}{3^n}

Using method of differences, one can prove that none of the above series is an integer.

Is it also true that ∑n=1∞n43n \displaystyle \sum_{n=1}^\infty \dfrac{n^4}{3^n} is not an integer?

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Pi Han Goh by Pi Han Goh
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