<s>[INST] $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ How to find the sum of the following series:
$\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$
Any hints.
 [/INST] The sum may be written as
$$\sum_{k=1}^{\infty} (-1)^{k+1} \frac{1\cdot 4 \cdots (3k-2)}{5^k k!} $$
Now, based on a suggestion from @Robert Israel:
$$(1+t)^{-1/3} = 1 -\frac1{1!}\frac13 t + \frac{1}{2!}\left ( -\frac13 \right ) \left ( -\frac{4}{3} \right )t^2 - \cdots$$
so that
$$1-(1+t)^{-1/3} = \frac1{1!}\frac13 t - \frac{1}{2!}\left ( -\frac13 \right ) \left ( -\frac{4}{3} \right )t^2 + \cdots $$
The series on the RHS reproduces the series in question when $t=3/5$, so that the sum is
$$1-\left ( \frac{8}{5} \right )^{-1/3} = 1-\frac12 5^{1/3}$$
</s>