This question is from my exam paper which i had appeared last semister i am trying to solve this but dont know the right answer.
Q: Test the convergence of the series
∞
∑ 3n/2(n+1)
n=1
Please replies with explanation if possible .......
This question is from my exam paper which i had appeared last semister i am trying to solve this but dont know the right answer.
Q: Test the convergence of the series
∞
∑ 3n/2(n+1)
n=1
Please replies with explanation if possible .......
There is probably one that is a little easier to use, but you can use the integral test here:
$\displaystyle \sum_{n=1}^{\infty}\frac{3n}{2\left(n+1\right)}\im plies\tfrac{3}{2}\int_1^{\infty}\frac{x}{x+1}\,dx$
Let $\displaystyle z=x+1\implies x=z-1$. Thus, $\displaystyle \,dz=\,dx$
Thus, the integral becomes $\displaystyle \tfrac{3}{2}\int_2^{\infty}\frac{z-1}{z}\,dz=\tfrac{3}{2}\int_2^{\infty}\left(z-\frac{1}{z}\right)\,dz$
This gives us $\displaystyle \tfrac{3}{2}\left.\left[\tfrac{1}{2}z^2-\ln\left|z\right|\right]\right|_2^\infty=\infty$
Thus, this diverges by the integral test.
-------------------------------------------------------------------------
I just thought of an easier way to do this...
You can show that the series is divergent by the nth term test (also called Divergence Test):
$\displaystyle \lim_{n\to\infty}\frac{3n}{2\left(n+1\right)}=\fra c{3}{2}\neq0$. Thus, the series diverges.