Verify-if-the-series-n-1-n-2n-5-n-2-3n-2-is-convergent-or-divergent-What-method-is-easier-

Question Number 97418 by Rio Michael last updated on 08/Jun/20

Verify if the series

\underset{n = 1}{\sum^{n}} \frac{2 n + 5}{n^{2} + 3 n + 2} is convergent or divergent .

What method is easier ?

Commented by Tony Lin last updated on 08/Jun/20

\underset{n = 1}{\sum^{\infty}} \frac{(n + 1) + (n + 2) + 2}{(n + 1) (n + 2)}

= \underset{n = 1}{\sum^{\infty}} \frac{1}{n + 2} + \underset{n = 1}{\sum^{\infty}} \frac{1}{n + 1} + 2 \underset{n = 1}{\sum^{\infty}} \frac{1}{(n + 1) (n + 2)}

= 2 \underset{n = 1}{\sum^{\infty}} \frac{1}{n} - \frac{1}{2} - 1 - \frac{1}{2} + 1

= 2 \underset{n = 1}{\sum^{\infty}} \frac{1}{n} - 1

\underset{n = 1}{\sum^{\infty}} \frac{1}{n} i s d i v e r g e n t

\Rightarrow \underset{n = 1}{\sum^{\infty}} \frac{2 n + 5}{n^{2} + 3 n + 2} i s d i v e r g e n t

Answered by mathmax by abdo last updated on 08/Jun/20

u_{n} = \frac{2 n + 5}{n^{2} + 3 n + 2} \Rightarrow u_{n} = \frac{n (2 + \frac{5}{n})}{n^{2} (1 + \frac{3}{n} + \frac{2}{n^{2}})} \Rightarrow u_{n} \sim \frac{2 + \frac{5}{n}}{n (1 + \frac{3}{n} + \frac{2}{n^{2}})} \sim \frac{2}{n}

Σ \frac{2}{n} diverges \Rightarrow Σ u_{n} diverges

Commented by Rio Michael last updated on 08/Jun/20

thank you sir, but sir do you have any other method ?