# Functional Series and Uniform Convergence Question

• Nov 28th 2010, 04:20 PM
garunas
Functional Series and Uniform Convergence Question
Let a $\displaystyle \in$ (0,1). Show that the functional series

$\displaystyle \displaystyle\sum_{j=0}^{\infty}(-t^2)^j$ where $\displaystyle t \in [-a,a]$

is uniformly convergent with the limit function

$\displaystyle f(t) = \frac{1}{1+t^2}$

I have absolutely no clue where to go from here.
Any help???
• Nov 28th 2010, 07:09 PM
Drexel28
Quote:

Originally Posted by garunas
Let a $\displaystyle \in$ (0,1). Show that the functional series

$\displaystyle \displaystyle\sum_{j=0}^{\infty}(-t^2)^j$ where $\displaystyle t \in [-a,a]$

is uniformly convergent with the limit function

$\displaystyle f(t) = \frac{1}{1+t^2}$

I have absolutely no clue where to go from here.
Any help???

Merely note that $\displaystyle |f_j(t)|=t^{2j}\leqslant (a)^{2j}$ for $\displaystyle t\in[-a,a]$, but since $\displaystyle \displaystyle \sum_{j=}^{\infty}a^{2j}$ converges, it follows by the Weierstrass M-test that $\displaystyle \displaystyle \sum_{j=0}^{\infty}f_j(t)$ converges uniformly on $\displaystyle [-a,a]$. To prove what it sums to merely note that $\displaystyle \displaystyle \frac{1}{1-\left(-t^2\right)}=\sum_{j=0}^{\infty}\left(-t^2\right)^j$