<s>[INST] Determining series formula Is there any procedure to follow when determining the function of a series? This seems simple but for I can't figure it out.

$$ \frac15 + \frac18 + \frac1{11} +\frac1{14} + \frac1{17}+\cdots$$

 [/INST] Generally speaking, when I see a series like this, I try to construct a generating function of some sort.  In this case:
$$f(x) = \sum_{k=1}^{\infty} \frac{x^{3 k+2}}{3 k+2}$$
You can easily show that
$$f'(x) = \frac{x^4}{1-x^3}$$
So we integrate and use $f(0)=0$ to evaluate $f(1)$.  In this case, it should be clear that this value will be infinite, but it may be instructive to see how the sum diverges.
We evaluate the integral by partial fractions:
$$\frac{x^4}{1-x^3} = x \left ( \frac{1}{1-x^3} - 1\right ) = \frac{1}{3} \left ( \frac{1}{1-x} + \frac{x+1/2}{(x+1/2)^2 + 3/4} - \frac{3/2}{(x+1/2)^2 + 3/4}\right)-x$$
Doing the integration and applying the initial condition reveals 
$$f(x) = \frac{1}{6} \log{(1+x+x^2)} - \frac{1}{\sqrt{3}} \arctan{\left[\frac{2 x+1}{\sqrt{3}}\right]} - \frac{1}{3} \log{(1-x)} - \frac{1}{2} x^2 + \frac{\pi}{6 \sqrt{3}}$$
So that, as $x \rightarrow 1^-$, $f(x)$ diverges logarithmically as one may expect.
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