Recall that the Maclaurin series is just another name for...

Recall that the Maclaurin series is just another name for the Taylor series at $x=0 .$ Each of the series in Exercises $31-34$ is the value of the Maclaurin series of a function $f(x)$ at some point. What function and what point? What is the sum of the series? $$ \frac{\pi}{3}-\frac{\pi^{3}}{3^{3} \cdot 3}+\frac{\pi^{5}}{3^{5} \cdot 5}-\cdots+\frac{(-1)^{k} \pi^{2 k+1}}{3^{2 k+1}(2 k+1)}+\cdots $$

Recall that the Maclaurin series is just another name for the Taylor series at $x=0 .$ Each of the series in Exercises $31-34$ is the value of the Maclaurin series of a function $f(x)$ at some point. What function and what point? What is the sum of the series?
$$
\frac{\pi}{3}-\frac{\pi^{3}}{3^{3} \cdot 3}+\frac{\pi^{5}}{3^{5} \cdot 5}-\cdots+\frac{(-1)^{k} \pi^{2 k+1}}{3^{2 k+1}(2 k+1)}+\cdots
$$

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Key Concepts

Function Identification via Series Comparison

Recognizing a series involves comparing its general term to known Maclaurin series expansions. When the series' structure—such as alternating signs, odd exponents, and denominator patterns—matches that of a well-known expansion like arctan x, one can determine both the function being represented and the specific value at which it is evaluated.

Maclaurin Series

A Maclaurin series is a type of Taylor series expansion where the function is expressed as an infinite sum of its derivatives evaluated at zero. This expansion represents the function as a power series in x and is a fundamental tool for approximating functions and analyzing their behavior near the origin.

Inverse Tangent Series

The Maclaurin series for the inverse tangent function, arctan x, is given by the series x - x³/3 + x?/5 - ... This alternating series with odd powers and odd-numbered denominators serves as a template for recognizing similar series patterns, enabling one to identify when a given series represents the arctan function evaluated at a specific point.

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