Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.7.58a
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 guidance1
Recall the given series expansion for \(e^x\):
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots\]
To find the series for \(\frac{d}{dx} e^x\), differentiate the series term-by-term. Use the power rule for differentiation:
\[\frac{d}{dx} x^n = n x^{n-1}\]
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.
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.
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 \(e^x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:58
Intro to Power Series
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.
Recommended video:
05:58Intro to Power Series
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.
Recommended video:
04:50Derivatives of General Exponential Functions
Related Practice
Textbook Question
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
a. Converges absolutely for x = 1
Textbook Question
Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)
f(x) = ln(cos x)
Textbook Question
Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)
f(x) = 1 / √(1 − x²)
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]