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?
Ya have,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.