Question

find the sum of the series
(1/2^k + 1/4^k)

Answer 1

The addition of geometric series. I figure you can just take the sum of each series individually and then add them?

Either way, since these are geo-series they fit the form of:
\[\sum_{i=0}^{\infty}ar^{n}\]A would be your starting point so to speak and r would be your ratio. In this case, your a is just 1 really and the r is 1/2. So you have:
\[1(\frac{ 1 }{ 2 })^k\]When you have something like this, the sum of the series is
\[\frac{ a }{ 1-r }\]So that being said, the first series just gives you:

\[\frac{ 1 }{ 1-\frac{ 1 }{ 2 } }= 2\]This then gets added to the other series which would just be:
\[\frac{ 1 }{ 1-\frac{ 1 }{ 4 } }= \frac{ 4 }{ 3 }\]So then its just the addition of these 2
\[2+\frac{ 4 }{ 3 }= \frac{ 10 }{ 3 }\]