Is the series $\sum_{1}^{\infty}\frac{6*11*16***(5n-4)(5n+1)}{8*13*18*21***(5n-2)(5n+3)}$ convergent or divergent

Question

$$\sum_{1}^{\infty}\frac{6\cdot 11\cdot 16\cdots(5n-4)(5n+1)}{8\cdot 13\cdot 18\cdot 21\cdots(5n-2)(5n+3)}$$

using ratio test I get 1.

$\sum_{1}^{\infty}\frac{6}{8\cdot 13\cdot 18\cdot 23\cdots (5n-2)(5n+3)}$==> can I you do integral test on this one, and by prooving that it divereges then the orginal function which is bigger is also divergent?

Answer 1

Your series is$$\sum_{n\ge1}\prod_{i=1}^n\frac{5i+1}{5i+3}\ge\sum_{n\ge1}\prod_{i=1}^n\frac{5i+1}{5i+6}=\sum_{n\ge1}\frac{6}{5n+6},$$which diverges by comparison with the harmonic series.