<s>[INST] How does this infinite series $1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots$ simplify to an integral $\int_0^1\frac{dx}{1+x^3}$? How does the infinite series below simplify to that integral?
$$1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots=\int_0^1\frac{dx}{1+x^3}$$
I thought of simplifying the series to the sum to infinity of $\frac{1}{6n-5} - \frac{1}{6n-2}$, but this did not help.
 [/INST] $$\int_{0}^{1}{\frac{dx}{1-(-x)^3}=\int_{0}^{1}{\sum_{n=0}^{\infty}{(-x)}^{3n}}}dx=\sum_{n=0}^{\infty}{(-1)^{3n}\int_{0}^{1}{x^{3n}}dx}$$
$$=\sum_{n=0}^{\infty}{\frac{(-1)^{3n}}{3n+1}}= 1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots $$
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