Power Series for arctan(x), Single Variable Calculus

(Taylor Series) In this lesson, we derive a power series representation for f(x) = arctan(x), centered at the origin, with interval of convergence [-1,1]. We visualize the convergence and use this power series to estimate the value of pi. [Note: It will turn out after we have learned the definition of a Taylor Series that the power series we derive here can more accurately be described as the "Taylor series for arctan(x) centered at 0," which can also be called the "Maclaurin Series for arctan(x)."]

To find this representation, we consider the derivative of arctan(𝑥), which is 1/(1+𝑥^2). Recognizing the similarity to the geometric series prototype 1/(1−𝑥), we rewrite 1/(1+𝑥^2) as 1/(1−(−𝑥^2)) and using integration to get to the power series for arctan(𝑥).

Observations: Its radius of convergence 𝑅=1 and interval of convergence [−1,1]. We note that all the powers of 𝑥 in the series are odd, reflecting the odd symmetry of arctan(𝑥): arctan(−𝑥)=−arctan(𝑥), for any value of 𝑥.

We plot partial sums of the power series to visualize convergence between 𝑥=−1 and 𝑥=1, noting the odd symmetry of the arctan function and its horizontal asymptotes at 𝑦=±𝜋/2.

Finally, we demonstrate a neat application of this power series in approximating pi. By setting 𝑥=1 in our power series, we derive:

𝜋 = 4(1−1/3+1/5−1/7+⋯)

This series allows us to approximate the irrational number 𝜋 using rational numbers!

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