\(\displaystyle \sum_{n=1}^{\infty}{\frac{1}{3n+2} \ = \ \frac{1}{5}+\frac{1}{8}+\frac{1}{11}\ +\frac{1}{14} \ + \ \frac{1}{17} \ + ...\)
\(\displaystyle The \ ratio \ test \ tells \ us \ nothing, \ limit \ = \ one, \ ergo, \ a \ different \ ploy \ must \ be \ utilize.\)
\(\displaystyle Let \ f(x) \ = \ \frac{1}{3x+2}, \ f'(x) \ = \ \frac{-3}{(3x+2)^{2}} \ and \ x \ \ge \ 1, \ hence \ a \ good \ candidate \ for \ the \ integral \ test.\)
\(\displaystyle Therefore, \ \lim_{b\to\infty}\int_{1}^{b}\frac{1}{3x+2} \ = \ \lim_{b\to\infty}\frac{ln|3x+2|}{3}\bigg]_{1}^{b} \ = \ \infty, \ hence\ the \ series \ diverges.\)
