<s>[INST] $1-\frac{1}{5\cdot 3^2}-\frac{1}{7\cdot 3^3}+\frac{1}{11\cdot 3^5}+\frac{1}{13\cdot 3^6}--++\cdots.$ I want to evaluate the series
$$1-\frac{1}{5\cdot 3^2}-\frac{1}{7\cdot 3^3}+\frac{1}{11\cdot 3^5}+\frac{1}{13\cdot 3^6}--++\cdots.$$
I can rewrite this as
$$1+\sum_{n\geq 1} (-3)^{-3n}\left(\frac{3}{6n-1}+\frac{1}{6n+1}\right)$$
The answer should be $\frac{\ln7}{2}$. Although I can't see explicitly, since I get a log function, maybe it can be solved by differentiating some test function $f(x)$ and then substituting an appropriate number.
 [/INST] For context and inspiration, let's remember that the linear factors in the power series for $\log(1+x)$ come from integrating a geometric series:
$$
\log(1+x) = \int_0^x \frac1{1+t}\,dt = \int_0^x \sum_{n=0}^\infty (-1)^n t^n \,dt = \sum_{n=0}^\infty (-1)^n \frac{t^{n+1}}{n+1}.
$$
So if we recognize our series as a special value of a power series, which turns out to be
$$
f(x) = 1-\sqrt3\biggl( \frac{x^5}5 + \frac{x^7}7 - \frac{x^{11}}{11} - \frac{x^{13}}{13} + \frac{x^{17}}{17} + \frac{x^{19}}{19} - \frac{x^{23}}{23} - \frac{x^{25}}{25} + \cdots \biggr)
$$
evaluated at $x=\frac1{\sqrt3}$, then we can work out what that power series is by writing it as an integral of a rational function:
\begin{align*}
f(x) &= 1 - \sqrt3 \int_0^x ( x^4 + x^6 - x^{10} - x^{12} + x^{16} + x^{18} - x^{22} - x^{24} + \cdots )\,dt \\
&= 1 - \sqrt3 \int_0^x (t^4+t^6)(1-t^6+t^{12}-t^{18}+\cdots)\,dt \\
&= 1 - \sqrt3 \int_0^x \frac{t^4+t^6}{1+t^6} \,dt = 1 - \sqrt3 \int_0^x \frac{t^4}{t^4-t^2+1} \,dt.
\end{align*}
Using partial fractions (well, telling my computer to use partial fractions), we obtain
\begin{align*}
f(x) &= 1 - \int_0^x \biggl( \sqrt3 + \frac{t-\sqrt3/2}{t^2-t\sqrt3+1} - \frac{t+\sqrt3/2}{t^2+t\sqrt3+1} \biggr) \,dt \\
&= 1 - x\sqrt3 - \frac{\log (x^2-x\sqrt3+1)}2 + \frac{\log (x^2+x\sqrt3+1)}2,
\end{align*}
and therefore the series we want is
\begin{align*}
f\biggl( \frac1{\sqrt3} \biggr) &= 1 - 1 - \frac{\log (1/3)}2 + \frac{\log (7/3)}2 = \frac{\log7}2
\end{align*}
as predicted.
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