Textbook Question
Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.
a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]
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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 10.7.36a
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 ]
Verified step by step guidance1
Identify the general term of the series: \[a_n = (\sqrt{n+1} - \sqrt{n})(x - 3)^n\].
To find the radius of convergence, consider the absolute value of the terms without the \((x-3)^n\) part, and apply the Root Test or Ratio Test. Here, the Ratio Test is more straightforward.
Compute the limit $\(L = \(\lim\)_{n \(\to\) \(\infty\)} \(\left\)| \(\frac{a_{n+1}\)}{a_n} \(\right\)| = \(\lim\)_{n \(\to\) \(\infty\)} \(\left\)| \(\frac{(\sqrt{n+2}\) - \(\sqrt{n+1}\))(x - 3)^{n+1}}{(\(\sqrt{n+1}\) - \(\sqrt{n}\))(x - 3)^n} \(\right\)| = \(\lim\)_{n \(\to\) \(\infty\)} \(\left\)| \(\frac{\sqrt{n+2}\) - \(\sqrt{n+1}\)}{\(\sqrt{n+1}\) - \(\sqrt{n}\)} \(\right\)| \(\cdot\) |x - 3|.\)
Simplify the ratio of the square root differences by rationalizing or using the conjugate to find the limit of \[\frac{\sqrt{n+2} - \sqrt{n+1}}{\sqrt{n+1} - \sqrt{n}}\] as \[n \to \infty\].
Set the limit \[L < 1\] to find the radius of convergence, then solve the inequality \[|x - 3| < R\] where \[R\] is the radius. Finally, check the endpoints \[x = 3 - R\] and \[x = 3 + R\] by substituting back into the original series to determine if the series converges or diverges at those points, thus finding the interval of convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radius of Convergence
The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It can be found using tests like the Ratio Test or Root Test applied to the general term of the series. This radius determines the interval on the x-axis where the series behaves well.
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07:36
Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series’ center (here, x = 3) and extends to the radius of convergence on both sides. Endpoints must be checked separately to determine if the series converges or diverges there.
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08:44Interval of Convergence
Behavior of the General Term (√(n+1) − √n)
The term (√(n+1) − √n) simplifies to a form involving 1/(√(n+1) + √n), which behaves like 1/(2√n) for large n. Understanding this helps analyze the series’ terms and apply convergence tests correctly, especially since it affects the growth rate of coefficients in the power series.
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04:45Geometric Sequences - General Formula
Related Practice
Textbook Question
(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.
a. ∑ (from n=2 to ∞) [1 / (n ln n)]
Textbook Question
The series
sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯
converges to sec x for −π/2 < x < π/2.
a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?
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) = 1 / √(1 − x²)
Textbook Question
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.