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hivesaeed4
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How do we get the power series of arctan using integration. Could someone explain this step by step as I'm quite confused about it.
hivesaeed4 said:How do we get the power series of arctan using integration. Could someone explain this step by step as I'm quite confused about it.
hivesaeed4 said:How do we get the power series of arctan using integration. Could someone explain this step by step as I'm quite confused about it.
The power series representation of arctan is:
arctan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ... + (-1)^n * (x^(2n+1))/(2n+1) + ...
The interval of convergence for the power series of arctan is -1 < x < 1, which means that the series will converge for any value of x within this interval and diverge for any value of x outside of this interval.
The accuracy of the power series representation of arctan depends on the value of x and the number of terms used in the series. Generally, the more terms used, the more accurate the approximation will be.
No, the power series representation of arctan is an infinite series and can only provide an approximation of the exact value of arctan(x). However, the more terms used in the series, the closer the approximation will be to the exact value.
The power series of arctan and the power series of tan are inverse functions of each other. This means that the power series of arctan can be obtained by replacing x with 1/t in the power series of tan, and vice versa.