Hey Soroban, here's one you may like. Or anyone else, by all means.
Find the sum of the infinite series:
splitting the last fraction by partial fractions, we find that the sum is actually:
Thus, we have:
After writing out a few terms, it does not take long to realize that the will cancel all the terms from that are
Thus we can see that:
Was that the way you did it galactus?
Here's something you may find interesting then.
You are probably familiar with Euler's Basel problem. You know
and on and on.
Well, Euler found a general formula for the sum of the reciprocals of the EVEN powers.
So if you wanted
You could use the formula and get
The is the even Bernoulli numbers. They are tied in here also. Isn't it cool how this stuff ties together.
The Bernoulli numbers are an interesting thing to research if you're unfamiliar with them. Just a thought.
I took an advanced calculus class last semester and the professor actually derived this.
i think he mentioned this formula as well
Well, Euler found a general formula for the sum of the reciprocals of the EVEN powers.
I plan on looking up Bernoulli numbers one day (i'm one of those procrastinators, so chances are i won't look them up today even if i said i would). I don't know anything about them other than the fact they are used in this formula.The Bernoulli numbers are an interesting thing to research if you're unfamiliar with them. Just a thought.
I can derive this with Complex Analysis if anyone is interested.
Let me just post a real cool infinite summation:
.
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Leonard Euler's immortal approach was not formal. His idea was that a function can be factored by its zero's like a polynomial. Though his idea was informal it attracted attention. The general theorem is known as the Weierstrauss Factorization Theorem.
Off Topic: How is "Weierstrauss" pronounced properly, without an Americanized accent. Because I hated how my professor pronounced his name, it is so....so Americanized, if you know what I mean.