Show that

converges to some real number.

The problem hints that I'm supposed to use the fact that

converges uniformly on [0,1], which I've already proved.

I can't figure out how to connect these two facts at all... any help?

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- Apr 7th 2010, 08:06 PMpaupsersHow to show this sequence converges?
Show that

converges to some real number.

The problem hints that I'm supposed to use the fact that

converges uniformly on [0,1], which I've already proved.

I can't figure out how to connect these two facts at all... any help? - Apr 7th 2010, 08:37 PMsimplependulum
- Apr 8th 2010, 03:06 AMLaurent
Let's use your hint.

Because of the uniform convergence, the sum is continuous on . Furthermore, you may integrate term by term on . Note also that . We get:

.

The important thing is that the right-hand side is finite.

The above shows (means) that has a finite limit as .

However, this partial sum can be rewritten as ... - Apr 8th 2010, 01:02 PMpaupsers
- Apr 8th 2010, 01:09 PMLaurent