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If you let $t=\cos(x)$ and if $q$ is a positive integer you find something which is quite close to the integral formulae of the modified Bessel functions of the first kind (have a look here). Using $... • 213k 1 vote ### Can someone help find methods to study the asymptotic behavior of these integrals as$\rho \to \infty$? Here are some coarse ideas. Note that$t^2-1>0$on$t > 1$and$1-t^2>0$on$|t|<1.$Escribe$e^{-\rho t} = e^{-\rho t/2} e^{-\rho t/2} \leq e^{-\rho/2} e^{-\rho t/2}$for$t > 1.$Then,... • 6,147 1 vote ### Doubts on asymptotic criterion for$\sum_{n=1}^{\infty}a_n=\sum_{n=1}^{\infty}n^{a}\tan^{-1}\bigg(\frac{1}{n^a}\bigg)-e^{1/n}$with$a>0$. First note that $$\arctan x = x - \frac{{x^3 }}{3} + \mathcal{O}(x^5 )\quad \text{ and }\quad e^x = 1 + x + \mathcal{O}(x^2 )$$ as$x\to 0. Thus, \begin{align*} a_n &= n^a \left( {\frac{1}{{n^... • 19.7k 1 vote Accepted ### Low convergence of a perturbation solution to a non-linear ODE Writing the solution as $$y(x) = \gamma(\epsilon) (1-x) \ \ \ \quad \mathrm(0)$$ we see that\gamma$must satisfy $$\gamma^3\frac{\epsilon}{40}+\frac{\gamma}{12} -1=0. \ \ \ \quad \mathrm(1)$$ ... • 1,834 1 vote Accepted ### Speed of Convergence for some series (double sum) We will show that$S_n=o(n^{-(\alpha-1)})$. We claim$$n^{(\alpha-1)}S_n \approx n^{(\alpha-1)} \sum_{i=1}^n i^{-\alpha}\left[(n-i+1)^{-(\alpha-1)}-n^{-(\alpha-1)}\right] =\sum_{i=1}^n i^{-\alpha}\... • 315 1 vote ### Asymptotic for a binomial sum I followed the same suggestion as @Maxim of summing over$r=n-k$. The main reason is that for large$n$, ie$n\gg a, 1/|x|$, the largest terms are with$k$close to$n$: having them for small$r\$ is ...
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