Summing a complex series

**BG5965** · July 28th 2009, 06:55 AM

Hi, here is a problem involving summing a series involving a complex equation.

Sum this series:

$a_1 + a_2 + ... + a_{99}$
where $a_n = \frac{1}{(n + 1)\sqrt{n} + n\sqrt{n + 1}}$ for $n = 1, 2, ..., 99$

I'm not exactly sure how to start this question.

Please help, BG

**Jester** · July 28th 2009, 07:06 AM

Originally Posted by BG5965

Hi, here is a problem involving summing a series involving a complex equation.

Sum this series:

$a_1 + a_2 + ... + a_{99}$
where $a_n = \frac{1}{(n + 1)\sqrt{n} + n\sqrt{n + 1}}$ for $n = 1, 2, ..., 99$

I'm not exactly sure how to start this question.

Please help, BG

Notice that if you multiply top and bottom by $(n + 1)\sqrt{n} - n\sqrt{n + 1}$ then

$a_n = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1)^2 n - n^2(n + 1) } = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1) n } = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$

so the sum becomes telescopic.

**BG5965** · July 28th 2009, 07:07 AM

Sorry, but what does telescopic mean, in terms of maths?

**Jester** · July 28th 2009, 07:10 AM

Originally Posted by BG5965

Sorry, but what does telescopic mean, in terms of maths?

If you write out the terms you'll see that most of the terms cancel. Only the first and last survive. This is what we mean as a telescopic series.

**rowe** · July 28th 2009, 01:38 PM

Originally Posted by Danny

Notice that if you multiply top and bottom by $(n + 1)\sqrt{n} - n\sqrt{n + 1}$ then

$a_n = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1)^2 n - n^2(n + 1)^2 } = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1) n } = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$

so the sum becomes telescopic.

Did you mean:

$a_n = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1)^2 n - n^2(n + 1) }$

in the first part? And if so, the next part makes sense... but how did you get from $\frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1) n }$ to $\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$ ...

EDIT: Ah, nm, got it. $\frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}}$

**Jester** · July 28th 2009, 02:22 PM

Originally Posted by rowe

Did you mean:

$a_n = \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1)^2 n - n^2(n + 1) }$

in the first part? And if so, the next part makes sense... but how did you get from $\frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1) n }$ to $\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$ ...

EDIT: Ah, nm, got it. $\frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}}$

I did - thanks for pointing out the typo.

**rowe** · July 29th 2009, 03:21 AM

Interestingly, what happens when this becomes an infinite series? That is,

$a = \sum_{n=1}^{\infty} \left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$

I'm guessing it gets closer and closer to 1 since $\lim_{x\to\infty }{{1}\over{\sqrt{x+1}}} = 0$ , so a = 1.

**BG5965** · September 1st 2009, 06:29 AM

So does that mean the answer is:

$\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{1+1}} + \frac{1}{\sqrt{99}} - \frac{1}{\sqrt{99+1}}<br />$

I'm a little confused...

**Defunkt** · September 1st 2009, 12:01 PM

Originally Posted by rowe

Interestingly, what happens when this becomes an infinite series? That is,

$a = \sum_{n=1}^{\infty} \left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$

I'm guessing it gets closer and closer to 1 since $\lim_{x\to\infty }{{1}\over{\sqrt{x+1}}} = 0$ , so a = 1.

Yes, since $S_n = 1-\frac{1}{\sqrt{n+1}}$

@BG: $\frac{1}{ \sqrt{1}} - \frac{1}{ \sqrt{1+1}} +$ $\frac{1}{ \sqrt{1+1}} - \frac{1}{ \sqrt{1+2}} + \frac{1}{ \sqrt{1+2}} - ... - \frac{1}{ \sqrt{99}} + \frac{1}{ \sqrt{99}} - \frac{1}{ \sqrt{1+99}} = ?$

**BG5965** · September 8th 2009, 07:00 AM

So its:

$\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{99+1}}$ ?

Also, how did you get from:

$\frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{ (n + 1) n }$ to $\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$ using $\frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}}$ ?

Thanks, BG

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