Homework Statement
Prove that the sequence {c_n} converges to c if and only if the sequence {c_n - c} converges to zero.
Homework Equations
The Attempt at a Solution
First I prove that the convergence of {c_n - c} to zero implies that {c_n} converges to c by (all limits take n to...
Hmm, of course. Why did I not see that. I guess I was to focused on obtaining a recursion formula for the coefficients, because I thought I always had to do that. I see now that I don't. Thanks a lot for the help!
My problem is that I don't see how to obtain a recusion formula for the coefficients. Even when trying to solve just the corresponding homogeneous equation x y'(x) - y(x) = 0 I end up with something like n a_n - a_n = 0 and I can't see what to do.
I've tried solving the equation xy'(x) - y(x) = x^2 Exp[x] using the power series method. I assume that y has the form:
y = \sum_{n=0}^{\infty} a_n x^n
Inserting this in the diff. eq. gives:
\sum_{n=0}^{\infty} n a_n x^n - \sum_{n=0}^{\infty} a_n x^n = x^2 e^x
Now, in the other...
You could start by rewriting the integrand:
\frac{x^3}{\sqrt{x^2+1}}=x^3 (x^2+1)^{-1/2} = [x^{-6} (x^2+1)]^{-1/2}
And the apply the integration by parts formula. That should make it easier.