Uniform Convergence Problem?

Question

First determine the largest set on which
F(x) = (sum from n=1 to infinity) [e^(2nx)/e^(n*x^2)](1-x) converges pointwise. Then find the exact value of (integral from 3 to 4) F(x)dx.

In the first part, i found the interval of convergence as
(minus inf., -2] U (2,inf.). In the second part, i couldn't prove that f n converges uniformly on [3,4]. Can you help me with this?

This is not an homework, i'm studying for the upcoming exam.

Answer 1

If I understood your problem, the factor (1-x) has nothing to do with the point convergence.

The condition is that exp(-x^2+2x) should be less than 1, ie

-x^2+2x <0 , or x(x-2) > 0, then x<0 or x>2

Then your set will be

(-inf, 0) U (2, inf)

About the uniform convergence:

The general term of the series is

a_n=exp(-x^2+2x)^n

If 3<= x <= 4

As exp(x) grows monotonically for every x, and -x^2 + 2x decreases for x > -1, a_n is a decreasing function of x for x in [3;4]

Then
a_n < exp((-3^2+2*3)n) = e^(-3n)

Therefore, the series

0< sum(1 to inf) [e^(-x^2+2x)]^n < sum(1,inf) q^n

where q=e^(-3).

The geometric series converges uniformly, then your series converges uniformly too (for a given eps you can use the same N you choose for the geom. series).

Answer 2

I find the integral converges on (-inf,0) U (2,inf) to the sum
(1 - x)/[ e^(x^2 - 2x) - 1]. This is because the series is geometric with ratio 1/e^(x^2 - 2x), and the ratio is < 1 when x is in the interval stated above.