<s>[INST] What function does this Taylor Series represent? What is the function $f$ who's Taylor series is $1 - \frac{x}{4} + \frac{x^2}{7} - \frac{x^3}{10} + \cdots$ ?
I need to find the value of the series $ \sum^{\infty}_{n = 0}a_n = 1 - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} +  \cdots$ by finding $\lim_{x\rightarrow1^-} \sum^{\infty}_{n = 0}a_n x^n$. (Abel Summability)
I already did this for another problem.  I was given the series $\sum^{\infty}_{n = 0}b_n = \frac{1}{2\cdot 1} - \frac{1}{3\cdot 2} + \frac{1}{4\cdot 3} - \frac{1}{5\cdot 4} + \cdots$
I showed that $\sum^{\infty}_{n = 0}b_n x^n = \frac{\ln(1+x)}{x^2}+\frac{\ln(1+x)}{x}-\frac{1}{x}\rightarrow 2\ln(2)-1$ as $x\rightarrow1^- \Rightarrow \sum^{\infty}_{n = 0}b_n = 2\ln(2)-1$.  (This checks out numerically, as well)
However, I'm having trouble finding the functional representation of $ \sum^{\infty}_{n = 0}a_n x^n$
 [/INST] If
$f(x)
=\sum_{k=0}^{\infty} a_k x^k
$,
then
$f_{n, j}(x)
=\sum_{k=0}^{\infty} a_{j+kn} x^{j+kn}
=\frac1{n}\sum_{i=0}^{n-1} w^{ij} f(w^i x)
$
where
$w = e^{2\pi i /n}
$.
This is known as
multisection of series.
Here is one of many
available discussions:
http://mathworld.wolfram.com/SeriesMultisection.html
Since
$f(x)
=\frac{-\ln(1-x)}{x}
=\sum_{k=0}^{\infty} \frac{x^k}{k+1}
$,
$\sum_{k=0}^{\infty} \frac{x^{3k}}{3k+1}
=f_{0, 3}(x)
=\frac1{3}\sum_{i=0}^{2}  f(w^i x)
$
where
$w = e^{2\pi i /3}
$.
Since you want
$g(x)
=\sum_{k=0}^{\infty} \frac{(-1)^kx^k}{3k+1}
$,
$g(x)
=f_{0, 3}((-x)^{1/3})
$.
</s>