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Chapter 9: Problem 57

Define a power series centered at \(c\).

Short Answer

Expert verified
A power series centered at \(c\) is a series of the form \(\sum a_n (x-c)^n\). Where \(a_n\) are the coefficients, x is a variable, n is a non-negative integer, and the series is centered at the point c.

Step by step solution

01

Understand Power Series

A power series is a series of the form: \(\sum a_n (x-c)^n\) where \(a_n\) represents the coefficients of the series, n is a non-negative integer, x is a variable, and c is the point at which the series is centered.
02

Explain Centred at c

The term 'centered at c' refers to the point in the variable space where the power series is focused. This means that the values of the variable x in the power series are relative to c. In the power series \(\sum a_n (x-c)^n\), c is the center of the series.

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Key Concepts

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

Series Convergence
When we talk about a power series, such as \(\sum a_n (x-c)^n\), one of the key concepts is series convergence. This refers to whether the series adds up to a finite value as more and more terms are included. Imagine adding drops to a small cup of water; if it overflows no matter how small each drop is, we say the series diverges. But if you can add drops indefinitely without overflowing, the series converges.

For power series, the convergence depends not just on the coefficients \(a_n\), but also on the value of \(x\) relative to the center \(c\). There will be certain values of \(x\) for which the series converges, and other values for which it doesn't. Understanding the range of \(x\) values for which the series converges鈥攊ts region of convergence鈥攊s essential for working with power series effectively.
Radius of Convergence
Closely tied to the concept of series convergence is the radius of convergence. This is the distance from the center \(c\) within which the power series converges. Picture a circle on a number line centered at \(c\); its radius is the max distance you can go in either direction from \(c\) before the series starts to diverge.

To calculate the radius of convergence, we often use the ratio test or the root test. These tests provide a way to evaluate the limit of \(a_{n+1}/a_n\) or the nth root of \(a_n\), respectively, as \(n\) goes to infinity. If this limit exists and is less than one, the series converges absolutely. The ratio or root helps us establish this critical radius. Once we know the radius, we have a powerful tool to assess the behavior of the series around its center point \(c\).
Analytic Functions
A function is called analytic at a point if it can be represented by a power series that converges to the function's value in some interval around that point. In other words, the function can be 'approximated' by adding up a sequence of terms involving powers of \(x-c\), where \(c\) is the point of interest.

Analytic functions have great significance in calculus and complex analysis because they're 'smooth' and behave nicely, meaning they can be differentiated and integrated as if they were simple polynomials. This makes them incredibly useful for solving a wide range of problems in engineering, physics, and other fields where understanding change and area is critical. The power series representation of an analytic function isn't just a handy tool; it's often the most succinct way to capture the essence of the function's behavior in the vicinity of point \(c\).

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Most popular questions from this chapter

Use the formula for the \(n\) th partial sum of a geometric series \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\) Annuities When an employee receives a paycheck at the end of each month, \(P\) dollars is invested in a retirement account. These deposits are made each month for \(t\) years and the account earns interest at the annual percentage rate \(r\). If the interest is compounded monthly, the amount \(A\) in the account at the end of \(t\) years is $$ \begin{aligned} A &=P+P\left(1+\frac{r}{12}\right)+\cdots+P\left(1+\frac{r}{12}\right)^{12 t-1} \\ &=P\left(\frac{12}{r}\right)\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right] \end{aligned} $$ If the interest is compounded continuously, the amount \(A\) in the account after \(t\) years is $$ \begin{aligned} A &=P+P e^{r / 12}+P e^{2 r / 12}+P e^{(12 t-1) r / 12} \\ &=\frac{P\left(e^{n}-1\right)}{e^{r / 12}-1} \end{aligned} $$ Verify the formulas for the sums given above.

Let \(\left\\{x_{n}\right\\}, n \geq 0\), be a sequence of nonzero real numbers such that \(x_{n}^{2}-x_{n-1} x_{n+1}=1\) for \(n=1,2,3, \ldots \ldots\) Prove that there exists a real number \(a\) such that \(x_{n+1}=a x_{n}-x_{n-1}\), for all \(n \geq 1\).

Investigation (a) Find the power series centered at 0 for the function \(f(x)=\frac{\ln \left(x^{2}+1\right)}{x^{2}}\) (b) Use a graphing utility to graph \(f\) and the eighth-degree Taylor polynomial \(P_{8}(x)\) for \(f\). (c) Complete the table, where \(F(x)=\int_{0}^{x} \frac{\ln \left(t^{2}+1\right)}{t^{2}} d t\) and \(G(x)=\int_{0}^{x} P_{8}(t) d t\) (d) Describe the relationship between the graphs of \(f\) and \(P_{8}\) and the results given in the table in part (c).

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Consider the sequence \(\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find lim \(a_{n}\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(|r|<1\), then \(\sum_{n=1}^{\infty} a r^{n}=\frac{a}{(1-r)}\).
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