# Thread: Summing a complex series

1. ## Summing a complex series

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.

2. 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.

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.

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

4. 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.

5. 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. $\frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}}$

6. 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.

7. 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.

8. 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}}
$

I'm a little confused...

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

10. 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