<s>[INST] How to find sum of this series? $\sum_{n=1}^{+\infty}(-1)^{n+1}\prod_{k=1}^n\frac{3k-2}{5k}$ How to find sum of this series?
$$\frac{1}{5}-\frac{1\cdot 4}{5\cdot 10}+\frac{1\cdot 4\cdot 7}{5\cdot 10\cdot 15}-\ldots = \sum_{n=1}^{+\infty}(-1)^{n+1}\prod_{k=1}^n\frac{3k-2}{5k}$$
Any hint is welcome.
 [/INST] The sum of the series is: $$1-\frac{1}{2}\sqrt[3]{5}.$$
As suggested by Abhishek Bakshi, this comes from considering the Taylor series of
$$ f(x) = \left(1+\frac{3}{5}x\right)^{1/3} $$
around $x=0$.
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