Integrals!

[ComradeRed]

Aggh...I can't think straight right now. I need help turning a summation into a definite integral.
The summation series limit is "25n", for a given constant n. The summation itself is n/(x^2) where x begins at 1.

I know this is simple, but I can't think straight. Sorry, and thanks for the help in advanced.

[SteveA]

1/(x^2) can be written as x^-2, and the integral of that is -x^-1 or -1/x. So the integral of n/(x^2) would be -n/x.

The definite integral is the same as the difference between 2 integrals evaluating the function at the start and end points. So you could take -1/x and evaluate it at both points and subtract the two. Because you're actually looking for -n/x, you'd then multiply the result by n. One point you gave was x=1 which we'll assume is the starting point and would be -1/1 or -1, times n would be -n, so that's one point, then plug in the end value for x, do the same and subtract the two.

But when you mention the summation series, I'm a bit confused assume you need the approximation of it from a summation of discrete points which leaves the question of what formula you're suppose to use for the approximation - trapezoidal, rectangular?

[SteveA]

Ok, I think I get it. Ignore the summation stuff. Integrate n/(x^2) over x to get -n/x. Then a definite integral needs a start and end point for x. You're point x=1 gives -n/1 or -n. So the second point would be -n/x, which subtracted from -n would be -n/x - -n or n-n/x. This could be also be rewritten as n(x-1)/x.

Double check the math though. It's been a while since I've used calculus