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:

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- May 24th 2006, 01:19 AMsweetuniform 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, 01:40 PMThePerfectHackerQuote:

Originally Posted by**sweet**

$\displaystyle \sum^{\infty}_{k=0}\frac{x}{(1+x^2)^k}$

Thus, $\displaystyle a_{k+1}=\frac{x}{(1+x^2)^{k+1}}$ and $\displaystyle a_k=\frac{x}{(1+x^2)^k}$ apply the ratio test,

$\displaystyle \lim_{k\to \infty}\frac{x}{(1+x^2)^{k+1}}\cdot\frac{(1+x^2)^k }{x}$

This gives,

$\displaystyle L=\frac{1}{1+x^2}$

It converges*absolutely*when,

$\displaystyle L<1$

Thus,

$\displaystyle 1+x^2>1$

Thus, $\displaystyle x>0$.

It diverges when,

$\displaystyle L>1$

thus,

$\displaystyle 1+x^2<1$

WHich is impossible.

It is inconclusive when, $\displaystyle L=0$,

Thus, $\displaystyle x=0$.

But we can this that this series is convergent abolsutely thus, this is absolutely convergenet everywhere. - May 24th 2006, 02:44 PMsweet
ok now we know it's convergence but what about uniform convergence