what-is-the-sum-of-1-14-1-35-1-65-1-104-

Question Number 95145 by i jagooll last updated on 23/May/20

what is the sum of

\frac{1}{14} + \frac{1}{35} + \frac{1}{65} + \frac{1}{104} + \dots ?

Answered by bobhans last updated on 23/May/20

S = \underset{n = 1}{\sum^{\infty}} (\frac{2}{(3 n + 1) (3 n + 2)}) = \frac{2}{3} \lim_{x \to \infty} \underset{n = 1}{\sum^{x}} (\frac{1}{3 n + 1} - \frac{1}{3 n + 4})

= \frac{2}{3} \lim_{x \to \infty} [\frac{1}{4} - \frac{1}{3 x + 4}] = \frac{2}{3} \times \frac{1}{4} = \frac{1}{6}

Commented by i jagooll last updated on 23/May/20

thank you