Evaluate the sum? problem.. I need to evaluate the sum: can someone help?
The series is as follows:
$$
\frac 14 - \frac {1}{2(4)^2} + \frac{1}{3(4)^3} - \cdots + \frac{(-1)^{(x+1)}}{x(4)^x}
$$

We know, $$\sum_{0\le s\le n-1}y^s=\frac{y^n-1}{y-1}$$ 
Integrating either sides wrt $y$, we get $$\sum_{1\le r\le n}\frac{y^r}r=\int \left(\frac{y^n-1}{y-1}\right) dy$$
If $\sum_{1\le r\le n}\frac{y^r}r=S(y,n),$
$\frac 14 - \frac {1}{2(4)^2} + \frac{1}{3(4)^3} - \cdots + \frac{(-1)^{(x+1)}}{x(4)^x}=-S(-\frac14, x+1)$
The formula for the general summation is explained here.
