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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 \(1\).

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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.

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Most popular questions from this chapter

The Legendre Equation. Problems 22 through 29 deal with the Legendre equation $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\alpha(\alpha+1) y=0 $$ As indicated in Example \(3,\) the point \(x=0\) is an ordinaty point of this equation, and the distance from the origin to the nearest zero of \(P(x)=1-x^{2}\) is 1 . Hence the radius of convergence of series solutions about \(x=0\) is at least 1 . Also notice that it is necessary to consider only \(\alpha>-1\) because if \(\alpha \leq-1\), then the substitution \(\alpha=-(1+\gamma)\) where \(\gamma \geq 0\) leads to the Legendre equation \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\gamma(\gamma+1) y=0\) It can be shown that the general formula for \(P_{n}(x)\) is $$ P_{n}(x)=\frac{1}{2^{n}} \sum_{k=0}^{\ln / 2-} \frac{(-1)^{k}(2 n-2 k) !}{k !(n-k) !(n-2 k) !} x^{n-2 k} $$ where \([n / 2]\) denotes the greatest integer less than or equal to \(n / 2 .\) By observing the form of \(P_{n}(x)\) for \(n\) even and \(n\) odd, show that \(P_{n}(-1)=(-1)^{n} .\)

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

The Legendre Equation. Problems 22 through 29 deal with the Legendre equation $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\alpha(\alpha+1) y=0 $$ As indicated in Example \(3,\) the point \(x=0\) is an ordinaty point of this equation, and the distance from the origin to the nearest zero of \(P(x)=1-x^{2}\) is 1 . Hence the radius of convergence of series solutions about \(x=0\) is at least 1 . Also notice that it is necessary to consider only \(\alpha>-1\) because if \(\alpha \leq-1\), then the substitution \(\alpha=-(1+\gamma)\) where \(\gamma \geq 0\) leads to the Legendre equation \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\gamma(\gamma+1) y=0\) Show that, if \(\alpha\) is zero or a positive even integer \(2 n,\) the series solution \(y_{1}\) reduces to a polynomial of degree \(2 n\) containing only even powers of \(x\). Find the polynomials corresponding to \(\alpha=0,2,\) and \(4 .\) Show that, if \(\alpha\) is a positive odd integer \(2 n+1,\) the series solution \(y_{2}\) reduces to a polynomial of degree \(2 n+1\) containing only odd powers of \(x .\) Find the polynomials corresponding to \(\alpha=1,3,\) and \(5 .\)

Find two linearly independent solutions of the Bessel equation of order \(\frac{3}{2}\), $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{9}{4}\right) y=0, \quad x>0 $$

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. \(x^{2} y^{\prime \prime}-x(x+3) y^{\prime}+(x+3) y=0\)

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