Is the series covergent or divergent?

[shaiqbashir]

is the series covergent or divergent?

I want to know that is the following series convergent or divergent??

$$\sum \frac{2}{\sqrt{n}+1}$$


when i apply divergent test to it, it comes equal to 0 , it means that divergent test gets failed. then how to solve it?

which test i should apply?

the correct answer in my copy is written as COnvergent, but I am not getting a convergent answer.

please help me as soon as possible!

Thanks in advance!

 

[StatusX]

Compare it to $\sum 1/n$.

 

[TD]

the correct answer in my copy is written as COnvergent, but I am not getting a convergent answer.

Are you sure about that? Perhaps you should specify the limits, or may we assume from 0 (or 1?) to infinity?

 

[shaiqbashir]

Thanks for ur Help StatusX!

I applied Limit comparison test and it works with series 1/n as Bn.

thanks a lot once again!

 

[TD]

But you know that the harmonic series (1/n) diverges, right?

 

[shaiqbashir]

okz okz! I am sorry for some mistakes.

the correct answer of the above series is this that the series is DIVERGENT not convergent. it was my mistake!
So that means that Status X was correct, we can compare it with 1/n and we can also compare it with 1/(n^1/2)

both will give divergence as result by using Basic Comparison test.

Now I am having another problem which looks like to me of the same sort. I don't know its correct answer. here is the problem :

$$\sum \frac{1}{\sqrt{n}+\sqrt{n+1}}$$

we have to find that is the following series converging or diverging??

here is what i have done:

The Divergence Test seems to be failed here as it is giving me an answer equals to zero!

Now if i go for basic comparison test, what series should i consider in order to compare it with above series?

I have worked on the following series and they don't give me proper results:

$$\sum \frac{1}{\sqrt{n}}$$

$$\sum \frac{1}{\sqrt{n+1}}$$

$$\sum \frac{1}{n}$$

$$\sum \frac{1}{n^(1.5)}$$

Then what should i consider?

please answer this question as soon as possible!

thanks in advance!

 

[StatusX]

What do the terms look like as n gets very large? That should give you an idea of what to compare it to.