How can I analyze for which $s$ the integral $\int_{0}^{1} \frac{x^s} {\sqrt[3] {1+x^7} - 1} dx$ converges? I simplified $\sqrt[3] {1+x^7}$ using the binomial formula but I don't know how to continue.
2026-03-28 02:42:46.1774665766
Integral convergence using Taylor expansion
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The simplest is using equivalents near $0$:
By the binomial formula, $\sqrt[3]{1+x^7}=1+\frac13 x^7+o(x^7)$, so the denominator is $$\sqrt[3]{1+x^7}-1=1+\frac13 x^7+o(x^7)-1= \frac13 x^7+o(x^7),$$ which shows that $\sqrt[3]{1+x^7}-1\sim_0\frac13 x^7$, hence $$\frac{x^s} {\sqrt[3] {1+x^7} - 1}\sim_0 3\mkern 1mu x^{s-7},$$ and the latter converges if and only if $\;s-7>-1$.