my friend has sent me this series problem
$\displaystyle\sum_{n=1}^\infty \frac{1\cdot4\cdot7\cdots(3n-2))}{5^n \cdot n!}$
And he showed me that it converges to$(\frac{5}{2})^{1/3} -1$Wich is$\displaystyle\sum_{n=1}^\infty \frac{1}{5^n \cdot n!} \displaystyle\prod_{k=1}^n 3k-2$
But when I write$\displaystyle\sum_{n=1}^\infty \displaystyle\prod_{k=1}^n \frac{3k-2}{5^n \cdot n!}$ I got a different result, but they must be the sameSince the term $5^n×n!$ is constant in $k$ so the multiplication must be commutative and the index Don't effects, but why does this happened?
