\(\dfrac14+\dfrac2{4^2}+\dfrac1{4^3}+\dfrac2{4^4}+\cdots\)
\(=\left(\dfrac14+\dfrac1{4^2}+\dfrac1{4^3}+\dfrac1{4^4}+\cdots\right)+\left(\dfrac1{4^2}+\dfrac1{4^4}+\cdots\right)\)
\(=\displaystyle\sum_{n=1}^\infty\left(\frac14\right)^n+\sum_{n=1}^\infty\left(\frac14\right)^{2n}\)
\(=\displaystyle\sum_{n=0}^\infty\left(\frac14\right)^n-1+\sum_{n=0}^\infty\left(\frac14\right)^{2n}-1\)
\(=\displaystyle\sum_{n=0}^\infty\left(\frac14\right)^n+\sum_{n=0}^\infty\left(\frac1{16}\right)^n-2\)
\(=\dfrac1{1-\frac14}+\dfrac1{1-\frac1{16}}-2\)
\(=\dfrac25\)
