Evaluating the series: $1 +(1/3)(1/4) +(1/5)(1/4^2)+(1/7)(1/4^3)+ \cdots$ 
Sum the series:
  $$1 + \dfrac13 \cdot \dfrac14 + \dfrac15 \cdot \dfrac1{4^2} + \dfrac17 \cdot \dfrac1{4^3} + \cdots$$

How can I solve this? I am totally stuck on this problem.

Let $$S = \sum_{n=0}^{\infty} \dfrac1{2n+1} \dfrac1{4^n}$$
Note that
$$\dfrac1{2n+1} = \int_0^1 x^{2n} dx$$
Hence, we can write
\begin{align}
S & = \sum_{n=0}^{\infty} \dfrac1{4^n}\int_0^1 x^{2n} dx = \int_0^1 \sum_{n=0}^{\infty} \left(\dfrac{x^2}{4} \right)^n dx\\
& = \int_0^1 \dfrac{dx}{1-\dfrac{x^2}4} = \int_0^1 \dfrac{dx}{2-x} + \int_0^1 \dfrac{dx}{2+x}\\
& = \left. -\log(2-x) \right \vert_0^1 + \left. \log(2+x) \right \vert_0^1 = \log 3
\end{align}
Hence,
$$\sum_{n=0}^{\infty} \dfrac1{2n+1} \dfrac1{4^n} = \log(3)$$
