Hello, I have a question:

Does it converge? If so, does it uniformly converge?

Thought of turning the ln into two different ln's, but I really can't find a way to prove a convergence of anything here...

Can you please help me?

Thanks :)

- June 25th 2010, 08:15 AMadam63a Series of functions : Does it converge? Does it uniformly converge?
Hello, I have a question:

Does it converge? If so, does it uniformly converge?

Thought of turning the ln into two different ln's, but I really can't find a way to prove a convergence of anything here...

Can you please help me?

Thanks :) - June 25th 2010, 08:30 AMAlso sprach ZarathustraI have...
for

The last term is monotonic decreasing so we can use the integral test... hmmm...

still thinking about this... - June 25th 2010, 12:15 PMchisigma
The term is a nonsense for n=1 , so that the series probably is...

(1)

The convergence of the 'infinite sum' (1) is equivalent to the convergence of the infinite product...

(2)

... and that is guaranted by the convergence of the series...

(3)

... which is true for any value of x. The series (3) converges uniformely in any finite interval of the x and the same holds for the series (1)...

Kind regards

- June 26th 2010, 05:21 AMadam63
Beautiful, thank you very much :)!

Only one little thing - how do you explain the "... and that is guaranted by the convergence of the series..." (from (2) to (3) )? - June 26th 2010, 07:04 AMchisigma
The convergence of the series...

(1)

... can be demostrated applying the 'integral test' considering that is...

(2)

Kind regards