Sum of given infinite series: $\frac14+\frac2{4 \cdot 7}+\frac3{4 \cdot 7 \cdot 10}+\frac4{4 \cdot 7 \cdot 10 \cdot 13 }+....$

Find the sum of infinite series

$$\frac{1}{4}+\frac{2}{4 \cdot 7}+\frac{3}{4 \cdot 7 \cdot 10}+\frac{4}{4 \cdot 7 \cdot 10 \cdot 13 }+....$$

Generally I do these questions by finding sum of $n$ terms and then putting $ \lim{n \to \infty}$ but here I am not able to find sum of $n$ terms. Could some suggest as how to proceed?

Solution 1:

Notice that

$$\frac k{\prod_{m=1}^k(3m+1)}=\frac1{3\prod_{m = 1}^{k-1} (3m+1)}-\frac{1}{3\prod_{m = 1}^k (3m+1)}$$

Which gives us a telescoping series:$$S_N=\frac{1}{3} - \frac{1}{3\prod_{m = 1}^N (3m+1)}$$

which tends to $1/3$ as suspected.

Solution 2:

The partial sums, according to Maple, are $$-{\frac {2\,{3}^{1/2-N}\pi}{27\,\Gamma \left( 4/3+N \right) \Gamma \left( 2/3 \right) }}+\frac{1}{3} $$ It should be possible to prove that by induction.