Homework Help: Determine if series is con/divergent

1. May 12, 2012

bfusco

1. The problem statement, all variables and given/known data

Ʃ[√(n+1)-√(n)]/√(n^2 +n)

3. The attempt at a solution

well i split the series up first.

Ʃ√(n+1)/√(n^2 +n) - Ʃ√(n)/√(n^2 +n)

next:

Ʃ[(n+1)/(n^2 +n)]^1/2 - Ʃ[n/(n^2 +n)]^1/2

at this point im not sure if i can take the limit and divide all the terms by the highest power of n.

2. May 12, 2012

Dick

Both of the series you've split it up into diverge. They are like 1/sqrt(n). I'd suggest you multiply numerator and denominator by sqrt(n+1)+sqrt(n) first. It's the quadratic conjugate trick. Then use a comparison test.

Last edited: May 12, 2012
3. May 13, 2012

bfusco

so im basically using the comparison test? im comparing these series to 1/n, which is divergent, and because these series are less than 1/n it also is divergent. also considering that 1 of the now 2 series is diveregent the whole thing is divergent.

4. May 13, 2012

Dick

The two series you've split it up into diverge. The original series does not diverge. Read my post again.

Last edited: May 13, 2012
5. May 13, 2012

sharks

If you split up the following fraction into its sum;
$$\sum^{\infty}_{n= ?}\frac {\sqrt{n+1}-\sqrt n}{\sqrt {n^2 +n}}=\sum^{\infty}_{n= ?}\frac {\sqrt{n+1}}{\sqrt {n^2 +n}}-\sum^{\infty}_{n= ?}\frac {\sqrt n}{\sqrt {n^2 +n}}$$
and then take the limits for both, you will get zero in each case, which by the nth-term test for divergence, doesn't say if both series converge or diverge.
Normally, isn't the multiplication of the conjugate done with the denominator, instead of the numerator as you suggested?

Last edited: May 13, 2012
6. May 13, 2012

LCKurtz

There isn't any "normally". You can rationalize either a numerator or denominator. What you do depends on your problem and what your goal with the particular problem is. Dick's post is spot on for a good suggestion.