Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Determine the radius of convergence of the given power series. $$ \sum_{n=1}^{\infty} \frac{\left(x-x_{0}\right)^{n}}{n} $$

Short Answer

Expert verified
Answer: The radius of convergence for the given power series is 1.

Step by step solution

01

Identify the terms of the series

The given power series is $$ \sum_{n=1}^{\infty} \frac{\left(x-x_{0}\right)^{n}}{n}. $$
02

Apply the Ratio Test

To apply the Ratio Test, we need to find the limit of the ratio of consecutive terms as \(n\) goes to infinity. We compute the absolute value of the ratio of the \((n+1)\)th term to the \(n\)th term: $$ \lim_{n\to\infty} \left|\frac{\frac{(x-x_0)^{n+1}}{n+1}}{\frac{(x-x_0)^n}{n}}\right|. $$
03

Simplify the limit

Simplify the limit expression by dividing the numerator by the denominator inside the absolute value: $$ \lim_{n\to\infty} \left|\frac{(x-x_0)^{n+1}n}{(x-x_0)^n (n+1)}\right|. $$ Then, simplify the powers of \((x-x_0)\): $$ \lim_{n\to\infty} \left|\frac{(x-x_0)n}{n+1}\right|. $$
04

Compute the limit

As \(n\) goes to infinity, the limit becomes: $$ \lim_{n\to\infty} \left|\frac{(x-x_0)n}{n+1}\right|=\left|x-x_0\right|. $$
05

Determine the radius of convergence

For the series to converge, the limit obtained in Step 4 must be less than 1. So, we have the inequality: $$ \left|x-x_0\right| < 1. $$ This inequality defines an interval of convergence centered at \(x_0\) with a radius of convergence equal to 1. Therefore, the radius of convergence for the given power series is \(1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Series
A power series is an infinite series of the form \[ \sum_{n=0}^{\infty} a_n (x - x_0)^n \.\] Here, \(a_n\) represents the coefficient of the nth term, \(x\) is the variable, and \(x_0\) is the center of the series. As a sum of infinitely many terms, power series can be thought of as polynomials with an infinite number of terms. These series are crucial because they can represent functions in areas of the complex plane where the function is differentiable.

Understanding the power series revolves around knowing where it converges, which means where the sum of its terms forms a finite value. The convergence is determined within a certain radius from the center \(x_0\), known as the radius of convergence. The interval within this radius is where the power series behaves nicely, meaning it converges to a function. Outside this interval, the series may diverge, meaning it does not settle to a finite sum.

In practice, to find the radius of convergence, tests like the Ratio Test are commonly used, which brings us to the next essential concept.
Ratio Test
The Ratio Test is a method used in calculus to determine the absolute convergence of an infinite series. Specifically, in the context of power series, it helps in identifying the radius of convergence. To apply the Ratio Test, you take the limit of the absolute value of the ratio of successive terms of the series.

The formal rule for the Ratio Test says that for a series \( \sum a_n\), if \[ \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L < 1 \.\] then the series converges absolutely. If \(L > 1\) or if the limit does not exist, then the series diverges. And if \(L = 1\), the test is inconclusive.

In our example, we utilize the Ratio Test to find the limit as \(n\) approaches infinity for the series' terms. The outcome of this limit, as per the Ratio Test, determines whether the series converges or diverges, and within what range of \(x\) values the convergence occurs. This range, or interval, is symmetric about \(x_0\) and defines the series' radius of convergence.
Limit of a Sequence
The concept of a limit in the context of a sequence is about what value the sequence approaches as the number of terms grows indefinitely large. Formally, the limit of a sequence \(a_n\) as \(n\) goes to infinity, denoted by \(\lim_{n \to \infty} a_n\), is the value that the terms of the sequence get closer to as \(n\) increases without bound.

In our exercise, we compute the limit of the ratio of the power series' consecutive terms to apply the Ratio Test. This limit is a way to assess how the terms of the series behave as they progress towards infinity, which is an essential step in determining the series' convergence.

In this specific example, as \(n\) becomes very large, the ratio of \(\frac{n}{n+1}\) approaches 1. Hence, we find that the limit of the absolute value of \(\left|\frac{(x-x_0)n}{n+1}\right|\) simplifies to \(\left|x-x_0\right|\). It means that as long as \(x\) is within 1 unit of \(x_0\), the terms of the power series will get increasingly closer to zero, which implies convergence within that interval, establishing a radius of convergence equal to 1.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the given differential equation has a regular singular point at \(x=0 .\) Determine the indicial equation, the recurrence relation, and the roots of the indicial equation. Find the series solution \((x>0)\) corresponding to the larger root. If the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also. \(x^{2} y^{\prime \prime}-x(x+3) y^{\prime}+(x+3) y=0\)

Find all singular points of the given equation and determine whether each one is regular or irregular. \((x \sin x) y^{\prime \prime}+3 y^{\prime}+x y=0\)

The Legendre equation of order \(\alpha\) is $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\alpha(\alpha+1) y=0 $$ The solution of this equation near the ordinary point \(x=0\) was discussed in Problems 22 and 23 of Section 5.3 . In Example 5 of Section 5.4 it was shown that \(x=\pm 1\) are regular singular points. Determine the indicial equation and its roots for the point \(x=1 .\) Find a series solution in powers of \(x-1\) for \(x-1>0 .\) Hint: Write \(1+x=2+(x-1)\) and \(x=1+(x-1) .\) Alternatively, make the change of variable \(x-1=t\) and determine a series solution in powers of \(t .\)

Find all the regular singular points of the given differential equation. Determine the indicial equation and the exponents at the singularity for each regular singular point. \(y^{\prime \prime}+4 x y^{\prime}+6 y=0\)

(a) Show that \(x=0\) is a regular singular point of the given differential equation. (b) Find the exponents at the singular point \(x=0\). (c) Find the first three nonzero terms in each of two linearly independent solutions about \(x=0 .\) \(x y^{\prime \prime}+y^{\prime}-y=0\)

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free