# Discuss the convergence of the series

• Jan 4th 2010, 07:34 PM
flintstone
Discuss the convergence of the series
I have tried the ratio test and the comparison test but still i am unable to get the result

Question : Discuss the convergence of the infinite series $\displaystyle \sum_{n=1}^{\infty} \frac{3n}{4(n+1)}$

Solution :

Ratio test :

$\displaystyle \lim_{n \to \infty} \frac{u_{n+1}}{u_n} = \lim_{n \to \infty} \frac{3(n+1)}{4(n+2)}. \frac{4(n+1)}{3n} = \lim_{n \to \infty} \frac{(n+1)^2}{(n+2)n}$ now dividing nu and de by $\displaystyle n^2$ we get

$\displaystyle \lim_{n \to \infty} \frac{1+\frac{2}{n}+\frac{1}{n^2}}{1+\frac{2}{n}} = 1 (Ratio test failed )$

Comparison test :

$\displaystyle \lim_{n \to \infty} \frac{u_n}{v_n} = \lim_{n \to \infty} \frac{3n}{4(n+1)}$

Here $\displaystyle u_n = \frac{3n}{4(n+1)} ; v_n = \frac{1}{n}$

$\displaystyle \lim_{n \to \infty } \frac{u_n}{v_n} = \lim_{n \to \infty } \frac{3}{4(n+1)}$ = $\displaystyle \frac{3}{4} \lim_{n \to \infty } \frac{3}{4(n+1)}$ dividing the nu and de by n weget

$\displaystyle \frac{3}{4} \lim_{n \to \infty } \frac{\frac{3}{n}}{4+\frac{1}{n}}$ = $\displaystyle \frac{3}{4} . 0$ = 0

What should i do !!!!!
• Jan 4th 2010, 07:49 PM
Drexel28
Quote:

Originally Posted by flintstone
I have tried the ratio test and the comparison test but still i am unable to get the result

Question : Discuss the convergence of the infinite series $\displaystyle \sum_{n=1}^{\infty} \frac{3n}{4(n+1)}$

[/COLOR][/I][/B]!!!!!

You have got to be kidding me? Try comparing the series to 1
• Jan 4th 2010, 07:56 PM
flintstone
Quote:

Originally Posted by Drexel28
You have got to be kidding me? Try comparing the series to 1

Sorry i didnt get u ......which test should i use !!!!!!!
• Jan 4th 2010, 08:00 PM
Drexel28
Quote:

Originally Posted by flintstone
Sorry i didnt get u ......which test should i use !!!!!!!

Try the limit comparison test.
• Jan 4th 2010, 08:13 PM
flintstone
Quote:

Originally Posted by Drexel28
Try the limit comparison test.

sir i have tried the limit comparison test .........please see my first post for the steps .....
• Jan 4th 2010, 08:16 PM
Drexel28
Quote:

Originally Posted by flintstone
sir i have tried the limit comparison test .........please see my first post for the steps .....

$\displaystyle \lim_{n\to\infty}\frac{\frac{3n}{4(n+1)}}{1}=\frac {3}{4}$

Look up limit test for divergence.
• Jan 4th 2010, 08:26 PM
flintstone
Quote:

Originally Posted by Drexel28
$\displaystyle \lim_{n\to\infty}\frac{\frac{3n}{4(n+1)}}{1}=\frac {3}{4}$

Look up limit test for divergence.

u have taken $\displaystyle u_n = \frac{3n}{4(n+1)}$ and $\displaystyle v_n = 1$ is that correct ??........are we allowed to take $\displaystyle v_n =1$(just 1)
• Jan 4th 2010, 08:28 PM
Drexel28
Quote:

Originally Posted by flintstone
u have taken $\displaystyle u_n = \frac{3n}{4(n+1)}$ and $\displaystyle v_n = 1$ is that correct ??........are we allowed to take $\displaystyle v_n =1$(just 1)

Surely.
• Jan 4th 2010, 08:29 PM
Prove It
What Drexel is saying is that in any series,

If $\displaystyle \lim_{n \to \infty}{a_n} \neq 0$,

then the series diverges.

Look at the $\displaystyle n^{th}$ term in your series.

$\displaystyle a_n = \frac{3n}{4(n + 1)}$

$\displaystyle \lim_{n \to \infty}a_n = \lim_{n \to \infty}\frac{3n}{4(n + 1)}$

$\displaystyle = \frac{3}{4}\lim_{n \to \infty}\frac{n}{n + 1}$

$\displaystyle = \frac{3}{4}\lim_{n \to \infty}\left(1 - \frac{1}{n + 1}\right)$

$\displaystyle = \frac{3}{4}\cdot 1$

$\displaystyle = \frac{3}{4}$.

Since this is not 0, the series diverges.