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Evaluate the indefinite integral as a power series. What is the radius of convergence?

$ \int \frac {\tan^{-1} x}{x} dx $

Power series of

$$\int \frac{\arctan (x)}{x} d x=C+\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1)^{2}}$$

[ Radius of convergence is 1]

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you value the indefinitely girl as a power serious and well, this. Then read this convergence. All right, so first we're gonna expand this as a power serious equals to one. The racks times our attendance axe the ex. So, actually, that we spend this parts. And that is ICO too. And from one to infinity X to the power of two and modest one over two M minus one tons once wanted power and minus one. This is Yeah, exactly. And we can simplify this and exchange the other of nd grow and some. So it's going to be minus one to power on minus one extra part two in minus two over to in minus one, the X. So we can continue to simplify this into girl, which going to become That's going to become okay, Wait. Extract when it's twenty pounds in minus one over two months. One and with integral extra power two and monies to the ex. So at last, Eco's too. At last, The indie group is going to be X to the power two months, one over two. M minus one. So he goes to over two minutes once where okay. And the readers. Convergence, Actually, we spent from here, and we know that Upton and that's it's freezing various for acting and X is very close one.

University of Illinois at Urbana-Champaign