Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = 2x/(1 + x²)²

Verified step by step guidance

1. Recall the known power series for the function \( \frac{1}{1 - t} = \sum_{n=0}^{\infty} t^n \) for \( |t| < 1 \). This is a geometric series centered at 0.
2. Recognize that \( \frac{1}{(1 + x^2)^2} \) can be related to the derivative of \( \frac{1}{1 + x^2} \). Start with \( \frac{1}{1 + x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n} \) for \( |x| < 1 \).
3. Differentiate the series term-by-term to find the series for \( \frac{d}{dx} \left( \frac{1}{1 + x^2} \right) = \frac{d}{dx} \left( \sum_{n=0}^{\infty} (-1)^n x^{2n} \right) \). This gives \( \frac{-2x}{(1 + x^2)^2} = \sum_{n=0}^{\infty} (-1)^n 2n x^{2n-1} \).
4. Multiply both sides of the differentiated series by \(-1\) to isolate \( \frac{2x}{(1 + x^2)^2} \), which matches the function \( f(x) \). So, \( f(x) = 2x/(1 + x^2)^2 = \sum_{n=0}^{\infty} (-1)^{n+1} 2n x^{2n-1} \).
5. State the interval of convergence for the power series, which remains \( |x| < 1 \) because the operations of differentiation and multiplication by \( x \) do not change the radius of convergence.

Key Concepts

Power Series Representation

A power series is an infinite sum of terms in the form a_n(x - c)^n, where c is the center of the series. Representing functions as power series allows approximation and analysis using polynomials. Finding a power series centered at 0 means expressing the function as a sum involving powers of x.

Known Power Series and Manipulation

Using standard power series expansions, such as for 1/(1 - x) or 1/(1 + x^2), helps derive new series by algebraic operations like differentiation, multiplication, or substitution. Recognizing and manipulating these known series is key to finding the series for more complex functions.

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 involves applying convergence tests, such as the ratio or root test, and is essential to understand where the series accurately represents the function.