Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is


eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞


Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.


f(x) = x²eˣ

Verified step by step guidance

1. Recall the power series expansion for the exponential function centered at 0: \[e^{x} = \sum_{k=0}^{\infty} \frac{x^{k}}{k!}\]
2. To find the power series for \[f(x) = x^{2} e^{x}\], multiply the entire series for \[e^{x}\] by \[x^{2}\]: \[f(x) = x^{2} \sum_{k=0}^{\infty} \frac{x^{k}}{k!} = \sum_{k=0}^{\infty} \frac{x^{k+2}}{k!}\]
3. Rewrite the series to express it in a standard power series form \[\sum_{n=0}^{\infty} a_{n} x^{n}\] by changing the index of summation. Let \[n = k + 2\], so when \[k=0\], \[n=2\]. Thus, \[f(x) = \sum_{n=2}^{\infty} \frac{x^{n}}{(n-2)!}\]
4. Identify the coefficients \[a_{n}\] of the power series: For \[n \geq 2\], \[a_{n} = \frac{1}{(n-2)!}\], and for \[n < 2\], \[a_{n} = 0\] since the series starts at \[n=2\].
5. Determine the interval of convergence. Since the original series for \[e^{x}\] converges for all real \[x\] (i.e., \[(-\infty, \infty)\]), multiplying by \[x^{2}\] does not change the radius or interval of convergence. Therefore, the interval of convergence for \[f(x)\] is also \[(-\infty, \infty)\].

Key Concepts

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of x, typically centered at a point (here, 0). Understanding how to write functions as power series allows approximation and analysis of functions using polynomials. For example, eˣ can be represented as ∑ₖ₌₀∞ (xᵏ)/k!.

Manipulation of Power Series

To find the power series of f(x) = x²eˣ, multiply the known series for eˣ by x² term-by-term. This involves shifting powers and coefficients accordingly. Mastery of series operations like multiplication by xⁿ and term-wise addition or differentiation is essential.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges to the function. For eˣ, the series converges for all real x (−∞, ∞). Multiplying by x² does not change this interval, so the resulting series also converges everywhere.