The series
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + ⋯
converges to eˣ for all x.
a. Find a series for (d/dx)eˣ. Do you get the series for eˣ? Explain your answer.

Verified step by step guidance

1. Recall the given series expansion for \(e^x\): \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots\]
2. To find the series for \(\frac{d}{dx} e^x\), differentiate the series term-by-term. Use the power rule for differentiation: \[\frac{d}{dx} x^n = n x^{n-1}\]
3. Apply the derivative to each term: - The derivative of the constant term \(1\) is \(0\). - The derivative of \(x\) is \(1\). - The derivative of \(\frac{x^2}{2!}\) is \(\frac{2 x^{1}}{2!}\). - The derivative of \(\frac{x^3}{3!}\) is \(\frac{3 x^{2}}{3!}\). - Continue similarly for higher powers.
4. Simplify each term after differentiation: For example, \(\frac{2 x^{1}}{2!} = \frac{2 x}{2} = x\), and \(\frac{3 x^{2}}{3!} = \frac{3 x^{2}}{6} = \frac{x^{2}}{2}\), and so on.
5. After simplifying, observe the resulting series and compare it to the original series for \(e^x\). Explain whether the differentiated series matches the original series and why this confirms the property that the derivative of \(e^x\) is \(e^x\).

Key Concepts

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of a variable, often centered at zero. For example, eˣ can be written as the sum of xⁿ/n! for n from 0 to infinity. Understanding this allows us to manipulate and differentiate functions term-by-term.

Term-by-Term Differentiation of Power Series

If a power series converges within an interval, it can be differentiated term-by-term within that interval. Differentiating each term xⁿ/n! yields n*xⁿ⁻¹/n! = xⁿ⁻¹/(n-1)!, which reconstructs the original series shifted by one index, showing the derivative series corresponds to the original function.

Derivative of the Exponential Function

The exponential function eˣ is unique because its derivative is itself, meaning d/dx(eˣ) = eˣ. This property is reflected in its power series, where differentiating term-by-term reproduces the same series, confirming the function’s self-derivative nature.