Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = tan ⁻¹ (x/2), a = 0

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Recall the Taylor series expansion for \( \tan^{-1}(x) \) centered at \( a = 0 \), which is given by the power series \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \).

Since the function is \( f(x) = \tan^{-1}\left( \frac{x}{2} \right) \), substitute \( \frac{x}{2} \) in place of \( x \) in the series expansion.

This substitution transforms the series into \( \sum_{n=0}^{\infty} (-1)^n \frac{\left( \frac{x}{2} \right)^{2n+1}}{2n+1} \).

Simplify the term \( \left( \frac{x}{2} \right)^{2n+1} \) as \( \frac{x^{2n+1}}{2^{2n+1}} \) to write the series explicitly in terms of powers of \( x \).

Write the final power series in summation notation as \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1) 2^{2n+1}} \), which represents the Taylor series of \( f(x) \) centered at \( 0 \).

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (here, a = 0). It approximates the function near that point using polynomial terms, allowing complex functions to be expressed as power series.

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Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this infinite sum, making it easier to write and analyze series expansions of functions like f(x) = arctan(x/2).

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Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges to the function. Determining this interval ensures the series accurately represents the function within that range, which is crucial for understanding where the Taylor series is valid.

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