Poly and logarithms

Calculus Level 4

n=113n,n=1n3n,n=1n23n,n=1n33n \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=1n43n \displaystyle \sum_{n=1}^\infty \dfrac{n^4}{3^n} is not an integer?

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