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

Rivka Thorpe 2021-08-22 Answered
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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Expert Answer

Lacey-May Snyder
Answered 2021-08-23 Author has 15546 answers

\(\displaystyle{1}+\frac{{z}}{{2}}+\frac{{z}^{{2}}}{{4}}+\frac{{z}^{{3}}}{{8}}+\) … (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 \(\displaystyle\frac{{2}}{{{2}-{z}}}\).

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