Continuity of a Sum on R

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

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Originally Posted by cp05

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

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

Tonio

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

Quote:

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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?

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Weierstrass M-test - Wikipedia, the free encyclopedia

Read with particular care the 9th line. (Nerd)

Tonio

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

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Originally Posted by cp05

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

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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