Convergence and Continuity of a Series with Cosine and Factorials

[gutnedawg]

Homework Statement

Let f(x)=$$\Sigma$$ [(3^(n) + cos(n))/n!]X^n

1. Prove that for every x in [-10,10] the sum converges
2. Show that for every $$\epsilon$$ >0 there's an N independant of x in [-10,10] such that
|f(x) - $$\Sigma$$ [(3^(n) + cos(n))/n!]X^n | < $$\epsilon$$
3. Use 2 together with the fact that polynomials are contiuous everywhere to show that f is continuous in [-10,10]

The Attempt at a Solution

1. Can I just say that this is less than 4^n/n! *X^n and that converges by an+1/an test and that since it converges for all X it'll converge for a subset of R?
2. I'm not sure what to do here
3. Since I don't fully understand 2 I'm not sure what to do here. I feel like the polynomials converging part relates to the X^n with a0,a1,a2... being the part in front

 

[hunt_mat]

1) What about the ratio test? This will give you an idea of the radius of convergence.
2) Use the M-test here, you're showing that the fucnction is uniformly continuous.
3) 2 saying that the series defined is the power series for f(x), how do you prove continuity?

 

[gutnedawg]

I'm not sure if I can use the M-test, is that Weierstrass approximation theorem?

 

[gutnedawg]

I think this might be a limit problem?

 

[gutnedawg]

can anyone help me further on this?

 

[gutnedawg]

alright I understand the problem a little more but I'm still running into problems with part 2

I cannot use uniform convergence rules to prove this problem