Find all $\alpha$ such that $\int_{0}^{1}\frac{\sqrt{e^2+x^2} - e^{\cos x}}{x^{\alpha}}$ converges. Can you give me a hint how to start? I have an idea to change smth with Taylor series, but is it alright to say, i.e $e^{\cos x}$ ~ $e - \frac{ex^2}{2}$ in this integral?
2026-04-01 08:09:01.1775030941
Find all $\alpha$ such that integral converges
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Applying L'Hopital's Rule twice you get $\lim_{x \to 0} \frac {\sqrt {e^{2}+x^{2}} -e^{\cos\, x}} {x^{2}}=\frac {e+1/e} 2$. The integral converges iff $\alpha <3$.