calculus

I get \frac{1}{3} \left(1+\ln 3 + \frac{\pi \sqrt 3}{6} \right ) by a somewhat illegitimate procedure :D

Sorry, the answer should be
\dfrac{\pi}{3\sqrt{3}}
In #2 I calculated the sum from n=1 onwards. So we get the result by adding 1/2 to it.

@L: I started with expansion of \frac{1}{1-x^3}

Then integrated w.r.t x between limits 0 and t

Then integrated w.r.t t between limits 0 and 1

I've made an error in calc, but a log term is likely to b there

@theprophet
You have indeed made a mistake in the calculations. That integration gives \dfrac{\pi}{3\sqrt{3}} and no log terms.

then its fine.

For us non-students a brief sketch of the solution is permitted leaving the rest as an easy exercise for the students :D

You must write that correctly. I meant

\lim_{t \rightarrow 1^-} \int_0^t \left(\int_0^x \frac{1}{1-x^3} \ dx \right) \ dt