/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Represent the function \(\frac{4... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 1

Represent the function \(\frac{4}{(1-10 x)}\) as a power series \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}\) \begin{tabular}{l} \(c_{0}=\) \\ \(c_{1}=\) \\ \(c_{2}=\) \\ \(c_{3}=\) \\ \(c_{4}=\) \\ \hline \end{tabular} Find the radius of convergence \(R=\)

Short Answer

Expert verified
The coefficients are: \(c_0 = 4\), \(c_1 = 40\), \(c_2 = 400\), \(c_3 = 4000\), \(c_4 = 40000\). The radius of convergence \(R\) is \(\frac{1}{10}\).

Step by step solution

01

Identify the Standard Form

The goal is to represent the function in a form suitable for a power series expansion. The standard form for a power series is \[\frac{1}{1 - r}. \] Examine the given function \(\frac{4}{1 - 10x}\). We can say \(r = 10x\).
02

Write the Function as a Geometric Series

The function \(\frac{1}{1 - r}\) can be expanded as a geometric series, \[\sum_{n=0}^{\infty} r^n. \] Substituting \(r = 10x\), we get \[\sum_{n=0}^{\infty} (10x)^n. \]
03

Multiply the Series by the Constant

The original function has a factor of 4 in the numerator. Therefore, we need to multiply the series by 4, giving us \[4 \sum_{n=0}^{\infty} (10x)^n. \]
04

Write the Resulting Series

Substituting in the series representation, we get \[4 \sum_{n=0}^{\infty} (10^n x^n) = \sum_{n=0}^{\infty} 4 \cdot 10^n x^n. \]
05

Determine the Coefficients

By comparing the resulting series term-by-term with \(f(x) = \sum_{n=0}^{\infty} c_n x^n\), we identify the coefficients: \[ \begin{aligned} & c_0 = 4 \ & c_1 = 40 \ & c_2 = 400 \ & c_3 = 4000 \ & c_4 = 40000 \end{aligned} \]
06

Find the Radius of Convergence

The series \(\sum_{n=0}^{\infty} (10x)^n\) converges when \[|10x| < 1 \ |x| < \frac{1}{10}. \] Therefore, the radius of convergence is \[ R = \frac{1}{10}. \]

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Key Concepts

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

Geometric Series
A geometric series is a series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. It can be simplified and expressed as \( \frac{a}{1 - r} \) if \( |r| < 1 \).

In our case, we need to represent \( \frac{4}{1 - 10x} \) as a geometric series. First, identify the common ratio \( r = 10x \).

The goal is to express this function in the form \( \frac{1}{1 - r} \), which is the standard form for a geometric series. By substituting \( r = 10x \), the function can be expanded into a geometric series, giving us \( \sum_{n=0}^{\infty} (10x)^n \). This is an essential building block for finding the power series representation.
Radius of Convergence
The radius of convergence determines the interval within which the power series converges to the function. For a geometric series \( \sum_{n=0}^{\infty} r^n \), it converges if \( |r| < 1 \).

In our power series, \( r = 10x \).

Thus, the series converges when \( |10x| < 1 \rightarrow |x| < \frac{1}{10} \). Therefore, the radius of convergence is \( R = \frac{1}{10} \).

This information is critical because it tells us in what range of \( x \) values the series accurately represents the function. Outside this radius, the series may diverge.
Series Coefficients
Series coefficients \( c_n \) in a power series \( \sum_{n=0}^{\infty} c_n x^n \) determine the specific form of the function.

In our example, we have expanded the function into \( 4 \sum_{n=0}^{\infty} (10x)^n \), which simplifies to \( \sum_{n=0}^{\infty} 4 \cdot 10^n x^n \).

Comparing this with the general power series form \( f(x) = \sum_{n=0}^{\infty} c_n x^n \), we identify the coefficients as:
  • \( c_0 = 4 \)
  • \( c_1 = 40 \)
  • \( c_2 = 400 \)
  • \( c_3 = 4000 \)
  • \( c_4 = 40000 \)


These coefficients are crucial for understanding the exact form of the series.
Standard Form
To simplify functions into power series, expressing them in a standard form is essential. The standard form for a geometric series is \( \frac{1}{1 - r} \).

Our task was to rewrite \( \frac{4}{1 - 10x} \) as a power series. First, note that \( 10x \) matches the standard geometric form. Hence, \( \frac{1}{1 - 10x} \) expands to \( \sum_{n=0}^{\infty} (10x)^n \).

Given the factor 4 in the numerator, we multiply the series by 4 to maintain equivalence. Thus, we get \( 4 \sum_{n=0}^{\infty} (10^n x^n) = \sum_{n=0}^{\infty} 4 \cdot 10^n x^n \).

Transforming the given function into this standard form allowed us to find its power series representation.

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Most popular questions from this chapter

Sequences have many applications in mathematics and the sciences. In a recent paper \(^{3}\) the authors write The incretin hormone glucagon-like peptide- 1 (GLP-1) is capable of ameliorating glucose-dependent insulin secretion in subjects with diabetes. However, its very short half-life (1.5-5 min) in plasma represents a major limitation for its use in the clinical setting. The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium. For this exercise, assume the half-life of GLP-1 is 5 minutes. So if \(A\) is the amount of GLP-1 in plasma at some time \(t,\) then only \(\frac{A}{2}\) of the hormone will be present after \(t+5\) minutes. Suppose \(A_{0}=100\) grams of the hormone are initially present in plasma. a. Let \(A_{1}\) be the amount of GLP-1 present after 5 minutes. Find the value of \(A_{1}\). b. Let \(A_{2}\) be the amount of GLP-1 present after 10 minutes. Find the value of \(A_{2}\). c. Let \(A_{3}\) be the amount of GLP-1 present after 15 minutes. Find the value of \(A_{3}\). d. Let \(A_{4}\) be the amount of GLP-1 present after 20 minutes. Find the value of \(A_{4}\). e. Let \(A_{n}\) be the amount of GLP-1 present after \(5 n\) minutes. Find a formula for \(A_{n}\). f. Does the sequence \(\left\\{A_{n}\right\\}\) converge or diverge? If the sequence converges, find its limit and explain why this value makes sense in the context of this problem. g. Determine the number of minutes it takes until the amount of GLP-1 in plasma is 1 gram.

Compute the value of the following improper integral. If it converges, enter its value. Enter infinity if it diverges to \(\infty\), and -infinity if it diverges to \(-\infty\). Otherwise, enter diverges. $$ \int_{1}^{\infty} \frac{3 d x}{x^{2}+1}= $$ Does the series \(\sum_{n=1}^{\infty} \frac{3}{n^{2}+1}\) converge or diverge? [Choose: converges | diverges to +infinity | diverges to -infinity | diverges]

We can use known Taylor series to obtain other Taylor series, and we explore that idea in this exercise, as a preview of work in the following section. a. Calculate the first four derivatives of \(\sin \left(x^{2}\right)\) and hence find the fourth order Taylor polynomial for \(\sin \left(x^{2}\right)\) centered at \(a=0\) b. Part (a) demonstrates the brute force approach to computing Taylor polynomials and series. Now we find an easier method that utilizes a known Taylor series. Recall that the Taylor series centered at 0 for \(f(x)=\sin (x)\) is $$ \sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k+1}}{(2 k+1) !} $$ i. Substitute \(x^{2}\) for \(x\) in the Taylor series \((8.5 .7) .\) Write out the first several terms and compare to your work in part (a). Explain why the substitution in this problem should give the Taylor series for \(\sin \left(x^{2}\right)\) centered at \(0 .\) ii. What should we expect the interval of convergence of the series for \(\sin \left(x^{2}\right)\) to be? Explain in detail.

For the following alternating series, \(\sum_{n=1}^{\infty} a_{n}=0.45-\frac{(0.45)^{3}}{3 !}+\frac{(0.45)^{5}}{5 !}-\frac{(0.45)^{7}}{7 !}+\ldots\) how many terms do you have to compute in order for your approximation (your partial sum) to be within 0.0000001 from the convergent value of that series?

Find the first four terms of the Taylor series for the function \(\cos (x)\) about the point \(a=\) \(-\pi / 4\). (Your answers should include the variable \(\mathrm{x}\) when appropriate.) \(\cos (x)=\) \(+\ldots\)
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