Question

how to find the partial sum of this series 1/2+1/2+1/4+1/4+1/8+1/8+1/16+1/16 . . .

Answer 1

Just rewrite it as 2(1/2+1/4+1/8+1/16+...) , so the partial sum is twice that of 1/2+1/4+1/8+1/16+...

Answer 2

Or... the series is just 1+1/2+1/4 +1/8 +....
Formula for partial sum of a geometric series:\[S_{n}=a \cdot \frac{ 1-r^{n+1} }{ 1-r }\]where a is the first term, and r is the factor every next term is multiplied with.

Here this is: \[S_{n}=\frac{ 1-\left( \frac{ 1 }{ 2 } \right)^{n+1} }{ 1-\frac{ 1 }{ 2 } }=2\left( 1-\left( \frac{ 1 }{ 2 } \right)^{n+1} \right)=2-\left( \frac{ 1 }{ 2 } \right)^n\]

Answer 3

BTW, the limit of the series is 2!