Clan x86

General Forums => Academic / School => Math and Other Problems => Topic started by: Ender on October 31, 2006, 08:27:42 pm

Title: [Semi-Solved] Calculus Abuse
Post by: Ender on October 31, 2006, 08:27:42 pm

Find a closed-form expression for (http://latex.sidoh.org/?render=%5Csum_%7Ba%3D1%7D%5En%20ab%5E%7Ba%7D)
(a) without calculus
(b) with calculus.

Feel free to do (a) and (b) out of order, or just post the solution to one of them. (b) is more fun =-)

Edit: I'll post the solution on Monday.

Title: Re: Calculus Abuse
Post by: Ender on November 07, 2006, 08:59:19 pm

Solution

I'm leaving an open window for further investigation, so I'm posting the reasoning. I'm also being lazy and don't have much time to type. Need to clean the bird cage.

(a)

You break it up into a bunch of geometric series and add them up. For example, 2b^2 can be broken up into b^2 + b^2, each in different arithmetic series.

(b)

Basically, you do some algebra and then integrate a factor of the expression to get it into a geometric series, solve for the geometric series, then differentiate it ;P.

(http://latex.sidoh.org/?render=%5C%5C%0D%0A%5Csum_%7Ba%3D1%7D%5E%7Bn%7D%20ab%5E%7Ba%7D%20%3D%20b%281%20%2B%202b%5E%7B1%7D%20%2B%203b%5E%7B2%7D...%20%2B%20%28n-1%29b%5E%7Bn-1%7D%29%0D%0A%5C%5C%0D%0A%5C%5C%0D%0Alet%5C%20u%20%3D%20%281%20%2B%202b%5E%7B1%7D%20%2B%203b%5E%7B2%7D...%20%2B%20%28n-1%29b%5E%7Bn-1%7D%29%0D%0A%5C%5C%0D%0A%5C%5C%0D%0A%5C%5B%20%5Cint%20u%5C%2Cdb%5C%5D%5C%20%3D%20%5Csum_%7Bi%3D0%7D%5E%7Bn-1%7D%20b%5E%7Bi%7D%20%3D%20%5Cfrac%7B1-%20b%5E%7Bn%7D%7D%7B1-b%7D%0D%0A%5C%5C%0D%0A%5C%5C%0D%0A%5Csum_%7Ba%3D1%7D%5E%7Bn%7D%20ab%5E%7Ba%7D%20%3D%20b%5C%20%5Cfrac%7Bd%7D%7Bdb%7D%20%5C%5B%20%5Cint%20u%5C%2Cdb%5C%5D%5C%20%0D%0A)