series of integrals Mathematica Wolfram

Question

Does somebody knows how to represent series of integrals in Mathematica? I would like to represent something like $I_n=\int_0^{\pi/4}\tan^{2n}xdx$ and prove convergence of it, but after searching docs, it only shows some very simple discrete sums

Answer 1

One can calculate the integral directly:

$$\begin{align}I_n&=\int_0^{\pi/4}\tan^{2n}(x)~\mathrm dx\\&=\int_0^{\pi/4}\tan^{2(n-1)}(x)\sec^2(x)-\tan^{2(n-1)}(x)~\mathrm dx\\&=\frac1{2n-1}\tan^{2n-1}(x)\bigg|_0^{\pi/4}-I_{n-1}\\&=\frac1{2n-1}-I_{n-1}\end{align}$$

Thus,

$$\begin{align}I_n&=\frac1{2n-1}-I_{n-1}\\&=\frac1{2n-1}-\frac1{2n-3}+I_{n-2}\\&=\dots\\&=(-1)^nI_0+\sum_{k=1}^n\frac{(-1)^{n+k}}{2k-1}\\&=(-1)^n\frac\pi4+\sum_{k=1}^n\frac{(-1)^{n+k}}{2k-1}\end{align}$$

Answer 2

If you simply plug that integral into Mathematica, it will give

Integrate[Tan[x]^(2n),{x,0,Pi/4}] = (PolyGamma[0, 3/4 + n/2] - PolyGamma[0, 3/4 + n/2])/4

The polygamma function of order 0 is the digamma function (often denoted $\psi(x)$), the logarithmic derivative of the gamma function. Mathematica will also give you the limit

Limit[(PolyGamma[0, 3/4 + n/2] - PolyGamma[0, 3/4 + n/2])/4,n->Infinity] = 0

This can be shown from the fact that $\psi(x) = \ln(x) + O(x^{-1})$, and so $$ \frac{1}{4}\left[\psi\left(\frac{n}{2} + \frac{3}{4}\right) - \psi\left(\frac{n}{2} + \frac{3}{4}\right)\right] = \frac{1}{4}\left[\ln\left(1 + \frac{2}{2n+1}\right) + O(n^{-1})\right] = O(n^{-1}) $$ Which goes to zero as $n\rightarrow \infty$.