Functional Series and Uniform Convergence Question

**garunas** · November 28th 2010, 04:20 PM

Let a $\in$ (0,1). Show that the functional series

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

is uniformly convergent with the limit function

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

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

**Drexel28** · November 28th 2010, 07:09 PM

Originally Posted by garunas

Let a $\in$ (0,1). Show that the functional series

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

is uniformly convergent with the limit function

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

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

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

Thread: Functional Series and Uniform Convergence Question

LinkBack

Thread Tools

Display

Functional Series and Uniform Convergence Question

Similar Math Help Forum Discussions

Sum of two Uniform Convergent Functional Series

Uniform Convergence of a series

uniform convergence series

uniform convergence of a series.

Uniform Convergence of Series

Search Tags