# uniform convergence

• May 24th 2006, 02:19 AM
sweet
uniform convergence
if we have

sum{n=0/to infinit} f_n (x)

such that n in N and x in [-1.1]

f_n =x/(1+x^2)^n

r this series confergence?is it uniform convergence?

:confused:
• May 24th 2006, 02:40 PM
ThePerfectHacker
Quote:

Originally Posted by sweet
if we have

sum{n=0/to infinit} f_n (x)

such that n in N and x in [-1.1]

f_n =x/(1+x^2)^n

r this series confergence?is it uniform convergence?

:confused:

Ya have,
$\sum^{\infty}_{k=0}\frac{x}{(1+x^2)^k}$
Thus, $a_{k+1}=\frac{x}{(1+x^2)^{k+1}}$ and $a_k=\frac{x}{(1+x^2)^k}$ apply the ratio test,
$\lim_{k\to \infty}\frac{x}{(1+x^2)^{k+1}}\cdot\frac{(1+x^2)^k }{x}$
This gives,
$L=\frac{1}{1+x^2}$
It converges absolutely when,
$L<1$
Thus,
$1+x^2>1$
Thus, $x>0$.

It diverges when,
$L>1$
thus,
$1+x^2<1$
WHich is impossible.

It is inconclusive when, $L=0$,
Thus, $x=0$.
But we can this that this series is convergent abolsutely thus, this is absolutely convergenet everywhere.
• May 24th 2006, 03:44 PM
sweet
ok now we know it's convergence but what about uniform convergence