- #1

- 13

- 0

## Homework Statement

Test if the infinite series converge or diverge.

## Homework Equations

[tex]

\sum_{n=1}^{\infty}\frac{4n+3}{n(n+1)(n+2)}

[/tex]

## The Attempt at a Solution

I tried Ratio test:

[tex]a_{n+1} = \frac{4n+7}{(n+1)(n+2)(n+3)}[/tex]

[tex]a_{n} = \frac{4n+3}{n(n+1)(n+2)}[/tex]

[tex]\left|\frac{a_{n+1}}{a_{n}}\right| = \frac{4n+7}{(n+1)(n+2)(n+3)} \times \frac{n(n+1)(n+2)}{4n+3}

= \frac{n(4+7n)}{(n+3)(4n+3)}

= \frac{4n^{2}+7n}{4n^{2}+15n+9}[/tex]

[tex]lim_{n\rightarrow\infty} \left|\frac{a_{n+1}}{a_{n}}\right| = lim_{n\rightarrow\infty} \frac{4+\frac{7}{n}}{4+\frac{15}{n}+\frac{9}{n^{2}}} = \frac{4}{4} = 1[/tex]

The answer is inconclusive, and I can't seem to think of any other test yet.

Anyone can help me with this?

I will much appreciate it. Thanks!

Last edited: