# Thread: Continuity of a Sum on R

1. ## Continuity of a Sum on R

Show SUM( (sin(nx)/n^2) * x^3) is continuous on R

2. Originally Posted by cp05
Show SUM( (sin(nx)/n^2) * x^3) is continuous on R

This is a series of functions which converges everywhere (Weierstrass M Test) times the function $x^3$ , thus...

Tonio

3. why does the fact that it converges everywhere...namely, it's uniformly convergent, show that (sin nx)/n^2 is continuous?

4. Originally Posted by cp05
why does the fact that it converges everywhere...namely, it's uniformly convergent, show that (sin nx)/n^2 is continuous?

Weierstrass M-test - Wikipedia, the free encyclopedia

Read with particular care the 9th line.

Tonio

5. I read the wikipedia line....but I don't see why it's continuous if it converges uniformly

6. Originally Posted by cp05
I read the wikipedia line....but I don't see why it's continuous if it converges uniformly

Because the limit function of continuous uniformly convergent functions is continuous...!! If you want you can prove this by yourself: it's pretty easy, or else you can google or check any decent analysis book. It's pretty standard stuff.

Tonio