Show that 1/(√1+√3)+1/(√5+√7)+1/(√9+√11) . . . + 1/(√9997+√9999) > 24.

I've no idea how to solve this. Please help. Ty.

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- Dec 14th 2012, 06:12 PMKalodaShow that the sum of the series is greater than 24
Show that 1/(√1+√3)+1/(√5+√7)+1/(√9+√11) . . . + 1/(√9997+√9999) > 24.

I've no idea how to solve this. Please help. Ty. - Dec 14th 2012, 10:00 PMchiroRe: Show that the sum of the series is greater than 24
Hey Kaloda.

Are you familiar with mathematical induction? - Dec 14th 2012, 10:01 PMabenderRe: Show that the sum of the series is greater than 24
Try taking advantage of conjugates. For example, the first term can be rewritten .

In each case, the denominator is 2, so we can factor out a half.

The series eventually becomes .

Does this look nicer? Good luck.

-AB - Dec 15th 2012, 02:26 AMKalodaRe: Show that the sum of the series is greater than 24
@chiro:

No. Mathematical Induction is supposed to be our next topic. - Dec 15th 2012, 06:56 AMKalodaRe: Show that the sum of the series is greater than 24
- Dec 15th 2012, 01:15 PMjakncokeRe: Show that the sum of the series is greater than 24
I've been racking my head all day to do this without induction, i'm curious if anyone has an answer. I took a look at trying to do this with continued fractions but i had no idea what i was doing, if only i was ramanujan.

- Dec 15th 2012, 05:37 PMKalodaRe: Show that the sum of the series is greater than 24
Can you show me the solution using Mathematical Induction?

I know the steps but I don't know how to apply it in cases that utilize summations. - Dec 15th 2012, 07:26 PMrichard1234Re: Show that the sum of the series is greater than 24
- Dec 15th 2012, 07:48 PMMarkFLRe: Show that the sum of the series is greater than 24
If I were going to use induction, I would state the hypothesis:

or equivalently:

where

base case :

true.

Consider:

Now, adding this to the hypothesis, we have:

We have derived from , thereby completing the proof by induction, and we may now state:

- Dec 15th 2012, 09:17 PMjakncokeRe: Show that the sum of the series is greater than 24
I don't know why i didn't think of this earlier.

But if you have then

take

then

So after computing the integral i got , and after evaluating i got

so