Power series of tan^-1(x), with radius & interval of convergence, calculus 2 tutorial

We will find the power series expansion of tan^-1(x), i.e. the arctan(x). We will do so by integrating the power series of 1/(1+x^2). Remember the radius of convergence stays the same when we integrate or differentiate a power series. HOWEVER, we must do more work to check the convergence at the endpoints of the interval of convergence.,

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[ΚωνσταντινοςΔημητριου-τε]

But who guarantees that the alernating series Σ [(-1)^n][x^(2n+1)]/(2n+1) converges to arctan(1) and arctan(-1) for x = +1, -1 in order to assume that arctan(x) can be described by that series on the boundaries of the interval?

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[lokeshverma]

actually if you use ratio test you will see that this infinite series always converge (x belong to real set )but does it converge to give arctanx ? i am saying that at -1 and 1 it is right to say series converges but how do we know that it converges to produce arctanx again?

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[johanrichter]

Checking the power series converges at the endpoints is easy, it is more tricky to check that it converges to arctan. Abel's theorem is what one would use here.