Question

How to find the sum of this infinite series

Answer 1

\[1+7r+r^{2}+7r^{3}+r^{4}+7r^{5}+r^{6}\]

Answer 2

You should know that any infinite converging geometric series has the sum,
\[\sum_{n=0}^\infty ar^n=\frac{a}{1-r},~~\text{provided that }|r|<1\]
What you must do for the given series is try to write it in such a way that you can determine the sum immediately, based off of the formula above.

Here's what you would do:
\[\begin{align*}1+7r+r^2+7r^3+r^4+7r^5+r^6\cdots&=\left(1+r^2+r^4+r^6+\cdots\right)+7\left(r+r^3+r^5+\cdots\right)\\
&=\sum_{n=0}^\infty r^{2n} +7\sum_{n=0}^\infty r^{2n+1}
\end{align*}\]
From here, try to find the sum of each series.

Answer 3

thank you thank, I just figured out this problem when you posted the solution, but thank you for your time!