# For which z’s does the infinite series converge? To what function does it conver

For which z’s does the infinite series converge? To what function does it converge?
$$\displaystyle{1}+\frac{{z}}{{2}}+\frac{{z}^{{2}}}{{4}}+\frac{{z}^{{3}}}{{8}}+\ldots$$

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Lacey-May Snyder

$$\displaystyle{1}+\frac{{z}}{{2}}+\frac{{z}^{{2}}}{{4}}+\frac{{z}^{{3}}}{{8}}+$$ … (geometric series)
So, $$\displaystyle=\frac{{1}}{{{1}-{\left(\frac{{z}}{{2}}\right)}}}=\frac{{1}}{{\frac{{{2}–{z}}}{{2}}}}$$
So, $$\displaystyle=\frac{{2}}{{{2}–{z}}}$$
This series converges if $$\displaystyle{\left|{z}\right|}{<}{2}$$ and it converges $$\displaystyle\frac{{2}}{{{2}-{z}}}$$.