<s>[INST] Infinite Series $\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$ How do I find the sum of the following infinite series:
$$\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$$
The series definitely seems to be convergent.
 [/INST] \begin{array}{l}
 s_0 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {\frac{{x^{3k + 3} }}{{3k + 3}}}  \Rightarrow s'_0 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {x^{3k + 2} }  = \frac{{x^2 }}{{1 - x^3 }} \\ 
 s_1 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {\frac{{x^{3k + 4} }}{{3k + 4}}}  \Rightarrow s'_1 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {x^{3k + 3} }  = \frac{{x^3 }}{{1 - x^3 }} \\ 
 s_2 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {\frac{{x^{3k + 2} }}{{3k + 2}}}  \Rightarrow s_2 ^\prime  \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {x^{3k + 1} }  = \frac{x}{{1 - x^3 }} \\ 
 s_0 \left( x \right) + s_1 \left( x \right) + s_2 \left( x \right) = \sum\limits_{x = 0}^{ + \infty } {\left( {\frac{{x^{3k + 3} }}{{3k + 3}} + \frac{{x^{3k + 4} }}{{3k + 4}} + \frac{{x^{3k + 2} }}{{3k + 2}}} \right)}  \\ 
 \frac{x}{{1 - x^3 }} + \frac{{x^3 }}{{1 - x^3 }} - \frac{{2x^2 }}{{1 - x^3 }} = \frac{{x^3  + x - 2x^2 }}{{1 - x^3 }} = \frac{{x\left( {x - 1} \right)}}{{1 + x + x^2 }} \\ 
 \end{array}
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