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Chapter 11: Problem 9

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{n^{2}+1}$$

Short Answer

Expert verified
The series converges by the Limit Comparison Test.

Step by step solution

01

Identify the nature of the series

The series in question is \( \sum_{n=1}^{\infty} \frac{\cos ^{2} n}{n^{2}+1} \). This is an infinite series with terms \( a_n = \frac{\cos^2 n}{n^2 + 1} \). The terms of the series are positive since both \( \cos^2 n \) and \( n^2 + 1 \) are non-negative for all \( n \).
02

Compare with a known convergent series

To determine convergence, we will compare \( a_n = \frac{\cos^2 n}{n^2+1} \) with the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is a p-series with \( p=2 \). This series \( ightarrow rac{1}{n^2} \) converges.
03

Apply the Limit Comparison Test

To apply the Limit Comparison Test, consider:\[L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{\cos^2 n}{n^2+1}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{\cos^2 n \cdot n^2}{n^2+1}\]As \( n \to \infty \), \( \cos^2 n \) is bounded, so \( \lim_{n \to \infty} \cos^2 n = K \) (where \( 0 \leq K < 1 \)) and the dominant terms in the numerator and denominator are respectively \( n^2 \). Thus,\[L = \lim_{n \to \infty} \frac{K \cdot n^2}{n^2+1} = K \]Since \( K \) is a finite positive number, the series \( \sum_{n=1}^{\infty} \frac{\cos^2 n}{n^2+1} \) behaves like the convergent series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) in the limit, and thus converges.

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Key Concepts

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

Limit Comparison Test
The Limit Comparison Test is a powerful method used to determine if an infinite series converges or diverges. It is particularly useful when a series does not fit the pattern of any basic known series. Here's the basic idea:
  • We have a series \( \sum a_n \) that we want to test for convergence.
  • We compare it with another series \( \sum b_n \) whose convergence is already known.
  • We compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
If \( L \) is a positive finite number, then both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together. This test works because it highlights the dominant behavior of series terms as \( n \) becomes very large, effectively sidestepping complexities caused by smaller order terms.
Convergence of Infinite Series
Determining the convergence of infinite series is a fundamental concept in mathematical analysis. Convergence describes whether the sum of an infinite series settles on a finite value. Here's a simple breakdown of how this works:
  • For a series \( \sum_{n=1}^{\infty} a_n \) to converge, the sequence of partial sums \( S_N = a_1 + a_2 + ... + a_N \) must tend towards a finite limit as \( N \rightarrow \infty \).
  • If the partial sums never settle on a number and instead grow without bound, the series diverges.
Understanding convergence is crucial because many processes in mathematics and applied sciences can be modeled by infinite series. A convergent series can represent functions, solve differential equations, and model phenomena that appear in physics and engineering.
P-Series
A p-series is a special type of infinite series labeled \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. The convergence of a p-series depends on the value of \( p \):
  • If \( p > 1 \), the series converges. For instance, \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a convergent series.
  • If \( p \leq 1 \), the series diverges. This includes the well-known harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which diverges.
The p-series is fundamental in mathematical series analysis because it provides a clear threshold for convergence and serves as a benchmark for comparison against more complex series using tests like the Limit Comparison Test. By comparing an unknown series to a p-series, we can often determine convergence characteristics reliably.

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Most popular questions from this chapter

(a) Show that the function $$f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$ is a solution of the differential equation $$f^{\prime}(x)=f(x)$$ (b) Show that \(f(x)=e^{x}\)

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=x^{-2}, \quad a=1, \quad n=2, \quad 0.9 \leqslant x \leqslant 1.1$$

Evaluate the indefinite integral as an infinite series. \(\int \frac{e^{x}-1}{x} d x\)

The Cantor set, named after the German mathematician Georg Cantor \((1845-1918),\) is constructed as follows. We start with the closed interval \([0,1]\) and remove the open interval \(\left(\frac{1}{3}, \frac{2}{3}\right) .\) That leaves the two intervals \(\left[0, \frac{1}{3}\right]\) and \(\left[\frac{2}{3}, 1\right]\) and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in \([0,1]\) after all those intervals have been removed. (a) Show that the total length of all the intervals that are removed is \(1 .\) Despite that, the Cantor set contains infinitely many numbers. Give examples of some numbers in the Cantor set. (b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center one-ninth of a square of side \(1,\) then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is \(1 .\) This implies that the Sierpinski carpet has area \(0 .\)

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=e^{x^{2}}, \quad a=0, \quad n=3, \quad 0 \leqslant x \leqslant 0.1$$
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