Question

Use the formula to evaluate the series 1/4 + 1/9 + 1/27 + 1/81.

Answer 1

Is the first term supposed to be 1/4, or did you actually mean 1/3?

I'll assume the latter. Write

$S=\dfrac13+\dfrac19+\dfrac1{27}+\dfrac1{81}$
$S=\dfrac13+\dfrac1{3^2}+\dfrac1{3^3}+\dfrac1{3^4}$
$\dfrac13S=\dfrac1{3^2}+\dfrac1{3^3}+\dfrac1{3^4}+\dfrac1{3^5}$
$\implies S-\dfrac13S=\dfrac13-\dfrac1{3^5}$
$\dfrac23S=\dfrac13\left(1-\dfrac1{3^4}\right)$
$S=\dfrac12\left(1-\dfrac1{81}\right)$
$S=\dfrac{40}{81}$