Is this alternative representation of $f(x)=xe^x$ as Maclaurin series correct?

The index of the second sum can start from $1$ because $0/0!=0/1=0$. Then, with reindexing, $$\sum_{n=1}^\infty\frac n{n!}x^n=\sum_{n=1}^\infty\frac 1{(n-1)!}x^n=\sum_{n=0}^\infty\frac{x^{n+1}}{n!}$$

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Sequences And Series

Power Series

Taylor Expansion

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