# Continuity of a Sum on R

• May 3rd 2010, 11:42 AM
cp05
Continuity of a Sum on R
Show SUM( (sin(nx)/n^2) * x^3) is continuous on R
• May 3rd 2010, 06:40 PM
tonio
Quote:

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 \$\displaystyle x^3\$ , thus...

Tonio
• May 3rd 2010, 07:06 PM
cp05
why does the fact that it converges everywhere...namely, it's uniformly convergent, show that (sin nx)/n^2 is continuous?
• May 3rd 2010, 07:18 PM
tonio
Quote:

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. (Nerd)

Tonio
• May 4th 2010, 08:36 AM
cp05
I read the wikipedia line....but I don't see why it's continuous if it converges uniformly
• May 4th 2010, 08:40 AM
tonio
Quote:

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