Emanuel Keith

## Answered question

2022-06-21

Finding the value of trignometric series ${\mathrm{\Sigma }}_{0}^{\mathrm{\infty }}\frac{\mathrm{cos}nx}{{3}^{n}}$

### Answer & Explanation

Bruno Hughes

Beginner2022-06-22Added 25 answers

Hint. One may write, for $x\in \mathbb{R}$,
$\frac{\mathrm{cos}nx}{{3}^{n}}=\text{Re}{\left(\frac{{e}^{ix}}{3}\right)}^{n}=\text{Re}\phantom{\rule{mediummathspace}{0ex}}{z}^{n}$
with $z=\frac{{e}^{ix}}{3}$ then one may recall that
$\sum _{n=0}^{\mathrm{\infty }}{z}^{n}=\frac{1}{1-z},\phantom{\rule{1em}{0ex}}|z|<1.$

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