Find the sum of the series.

$\displaystyle \sum_{k = 0}^\infty \frac{3^{k-1}}{4^{3k+1}}$

$\displaystyle \sum_{k = 0}^\infty \frac{3^{k}*3^{-1}}{4^{3k}*4^{1}}$

$\displaystyle \sum_{k = 0}^\infty \frac{3^{k}}{(4^{3})^{k}*4^{1}*3^{1}}$

$\displaystyle (\frac{1}{12})\sum_{k = 0}^\infty \frac{3^{k}}{64^{k}}$

$\displaystyle (\frac{1}{12})\sum_{k = 0}^\infty (\frac{3}{64})^{k}$

So it's a geometric series with $\displaystyle a = 1$ and $\displaystyle r = 3/64$. So $\displaystyle \frac{1}{12}*\frac{1}{1 - \frac{3}{64}}$ = $\displaystyle \frac{16}{183}$ ?

The solutions manual had $\displaystyle \frac{3}{15,616}$. Am I doing this right?

Re: Find the sum of the series.

What you have done looks fine. Does your solutions manual post their full solution or just the answer?

Re: Find the sum of the series.

There was a full solution, but I only copied down $\displaystyle \frac{3}{15,616}$. I may have copied down the wrong answer... and I won't have a chance to check until tomorrow.

Re: Find the sum of the series.

Well, the solutions manual had:

$\displaystyle \sum_{k = 2}^\infty \frac{3^{k-1}}{4^{3k+1}}$

$\displaystyle \sum_{k = 0}^\infty \frac{3^{k+1}}{4^{3k+7}}$

$\displaystyle (\frac{3}{4^7})\sum_{k = 0}^\infty (\frac{3}{4^{3}})^k$

$\displaystyle \frac{3}{4^7}*\frac{1}{1 - \frac{3}{4^3}}$

$\displaystyle \frac{3}{15616}$.

But $\displaystyle \sum_{k = 2}^\infty \frac{3^{k-1}}{4^{3k+1}}$ was not the original problem, $\displaystyle \sum_{k = 0}^\infty \frac{3^{k-1}}{4^{3k+1}}$ was. So, pretty safe to say that's a typo?