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Calculate $B_a(E)$
The set $E$ is the set of all real numbers $x$ for which the series $\sum_{n=1}^{\infty} \frac{1}{3^n}$ converges. Describe $B_a(E)$ where $a=0$ and $a=\frac{1}{2}$.
I’m not really sure how to approach this. I know $a$ would give me the radius of convergence of the series, and I know it converges, but how do I use that to calculate $B_a(E)$?
A:
Your knowledge of the definition of the balls is good.
With $r=\frac{1}{2}$, we have $$\forall x\in E,\,\left|x-\frac{1}{2}\right|=\frac{1}{2}-\left(\frac{1}{2}-x\right)=-x-\frac{1}{2}=-x-\frac{1}{2}-\left|x\right|=x+\left|x\right|$$
and you can do then the following calculations:
$\begin{array}{|l|l|}\hline&\begin{cases}a=0\\x\in E\end{cases}\quad&\begin{cases}a=1\\x\in E\end{cases}\\\hline\\
x\in E&0\leqslant\left|x\right|\le
https://documenter.getpostman.com/view/21827487/UzXUPDuY
https://ello.co/caudestinc_n/post/tkt4dbbb9jecd-vhv4mnow
https://colab.research.google.com/drive/1bqM6ouPp_AhmWMZvlE0YRvZ0qj-X-AD4
https://documenter.getpostman.com/view/21881841/UzdwV7BT
https://documenter.getpostman.com/view/21882614/UzdwW7Yk
https://ello.co/0nendefnasko/post/ogagdjcje5lmmk0dr7tlpa
https://colab.research.google.com/drive/1KgIF2VOgNSIv-wcCJlI4UG0G5ewP-CoD
https://documenter.getpostman.com/view/21827769/Uzdv1nnY
https://documenter.getpostman.com/view/21909682/UzXYprCF
https://colab.research.google.com/drive/1kRS60vjA9EwPP_9M6RZuy1gDCuafD94x
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