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- Nov 29th 2008, 07:20 AMHarriganCalculus Homework, Tutor couldn't help, I've hit the wall
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- Nov 29th 2008, 09:16 AMgalactus
The sum of the 6th power identities, huh?. Well, here is my way. Years ago

when I figured this out, I thought I had done something. But, it would be silly

to think it was original, afterall.

Consider

Notice the coefficients?. They look binomial, don't they?

Now, plug in the respective identities for the powers. We must know them dong it this way.

Now, solve for

It is lotsa horrible algebra, but if done correctly you should get:

I am sure there is a simpler way, but you can do any of them this way as long as you know the previous ones. Just plod through the algebra. - Nov 29th 2008, 10:05 AMgalactus

Quote:

The series of reciprocals of primes

Quote:

. Here is a beautiful result proved by Euler in the year 1773. The result states that the series of all the reciprocals of primes diverges.

(1/2) + (1/3) + (1/5) + (1/7) + (1/11) + (1/13) ...

Your task is to prove this result. This is a bit of a tricky problem, so I will give you more elaborate hints. Assume to the contrary that the above series converges and arrive at a contradiction using the steps below. (Notation: p will denote a prime number)

This can probably be found on line somewhere. The outline of Euler's proof

of this can be found in "Euler, the Master of Us All" by William Dunham.