I have to approximate this integral with an error lesser than 0.1 using Taylor Series. This is the integral: $$\int_0^1 \arctan\left(\frac{1}{x^{10}}\right) \,\Bbb dx.$$ If I understood correctly, I have to determinate the Taylor Series expansion in terms of a series with alternating signs. I have to check if the series has a uniform convergence in [0,1] and then take the series sign outside the integral sign and integrate the $x$-dependent part. Once done, I have to estimate the error using Leibniz: $|S - S_{n+1}|<|a_n|$. To find it I just have to calculate $n$ terms and when I’ll find $a_n<0.1$ I can computer the series from 0 to $n-1$. Am I correct?
I think I figured out how to do this job but I really do not know how to determinate the Taylor series expansion… I tried to use the McLaurin’s $\arctan(x)$ where $x_0=0$ but the integrand function is not defined for $x=0$… how can I handle this kind of problem? Please, can someone explain me if everything I said is correct and explain me how to determinate the series expansion in this case?
Sorry for my bad English and thank you in advance for your help!