To evaluate the sum $\frac{1}{5}-\frac{1 \cdot 4}{5 \cdot 10}+\frac{1 \cdot 4 \cdot 7}{5 \cdot 10 \cdot 15}-\ldots$ Right now I am working through archived papers of a math aptitude quiz. For some reason I seem to be haveing a hard time with these series problems. I have managed to write the above series in a compact form but thats as far as I got. 
$$\frac{1}{5}-\frac{1 \cdot 4}{5 \cdot 10}+\frac{1 \cdot 4 \cdot 7}{5 \cdot 10 \cdot 15}-\ldots = \sum\limits_{i=0}^\infty (-1)^i\frac{\prod\limits_{j=0}^i (3j+1)}{5^{i+1}(i+1)!}$$
Help please!

Lab Bhattacharjee is right. This is just an instance of the usual binomial series. More precisely,
$$
(1+x)^{-1/3}=1-\frac13x+\frac{1\cdot4}{3^2\cdot 2!}x^2-\frac{1\cdot4\cdot7}{3^3\cdot 3!}x^3+\cdots,
$$
that converges and has the prescribed sum, whenever $-1<x< 1$.
Plugging in $x=3/5$ gives 
$$\frac12\root 3\of5=1-\frac15+\frac{1\cdot4}{5\cdot10}-\frac{1\cdot4\cdot7}{5\cdot10\cdot15}+\cdots.$$ 
This yields $1-\dfrac12\root3\of5$ as the value of the sum.
