Author Topic: Quiz4-problemSixE  (Read 263 times)

Xuefen luo

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Quiz4-problemSixE
« on: October 24, 2020, 02:50:58 AM »
Problem: Find a "Closed form" for the given power series:
\begin{align*} \sum_{n=1}^{\infty} z^{3n}.\end{align*}
Answer:
Since we know that
\begin{align*} \sum_{n=0}^{\infty} z^{n} &= \frac{1}{1-z}\\
\sum_{n=0}^{\infty} (z^3)^n &= \frac{1}{1-z^3}\end{align*}
Thus,
\begin{align*}
 \sum_{n=1}^{\infty} z^{3n} &=  \sum_{n=0}^{\infty} z^{3n} - z^{3*0}\\
 &=\sum_{n=0}^{\infty} z^{3n} - 1\\
 &= \frac{1}{1-z^3} - 1 \\
\end{align*}
Therefore, the closed form for $ \sum_{n=1}^{\infty} z^{3n}$  is $\frac{1}{1-z^3} - 1 $.