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Chapter 9: Problem 4

Determine whether each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}$$

Short Answer

Expert verified
The series converges.

Step by step solution

01

Understand the problem

We need to determine if the series \( \sum_{n=1}^{\infty} \frac{1}{n^2+1} \) converges or diverges. Evaluating the convergence of a series involves analyzing its terms and applying a convergence test.
02

Choose a convergence test

One common approach is to use the Comparison Test or the Limit Comparison Test. These involve comparing the series to a known benchmark series to determine convergence or divergence.
03

Select a comparison series

For the series \( \sum_{n=1}^{\infty} \frac{1}{n^2+1} \), a good series for comparison is \( \sum_{n=1}^{\infty} \frac{1}{n^2} \). The series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is known to converge (p-series with \( p = 2 \)).
04

Apply the Comparison Test

Notice that \( \frac{1}{n^2+1} < \frac{1}{n^2} \) for all \( n \geq 1 \) because adding 1 to \( n^2 \) increases the denominator, making the fraction smaller. Since \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, and \( \frac{1}{n^2+1} < \frac{1}{n^2} \), the Comparison Test tells us that \( \sum_{n=1}^{\infty} \frac{1}{n^2+1} \) converges.

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Key Concepts

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

Comparison Test
The Comparison Test is a handy tool in the world of mathematical series for deciding convergence. It works by comparing the series of interest to another series with known behavior.

Here鈥檚 how it works:
  • You select a series, say \( \sum a_n \), whose convergence is in question.
  • Find another series, \( \sum b_n \), with known convergence or divergence behavior.
  • If \( a_n \leq b_n \) for all terms \( n \), and if \( \sum b_n \) converges, then \( \sum a_n \) also converges.
  • Conversely, if \( a_n \geq b_n \) for all \( n \), and \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.
This test works because terms in a series give a "weight" to its convergence behavior. By being smaller and decreasing, the terms in a_n wouldn鈥檛 outweigh those in a known convergent series, ensuring its convergence as well.
Limit Comparison Test
The Limit Comparison Test can be seen as a refined cousin of the Comparison Test. It's another strategy to check series convergence but uses limits to do so.

Here's the process:
  • Take the series \( \sum a_n \) that you want to test.
  • Choose another series \( \sum b_n \) that's simpler but closely resembles \( \sum a_n \).
  • Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
  • If \( 0 < L < \infty \), both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together.
This test is brilliant because it provides a ratio which acts as a bridge, connecting the behavior of two series. It's like finding a balance point, revealing that the small-term behavior mirrors the known series behavior.
p-series
A p-series is a special type of infinite series in the form \( \sum_{n=1}^{fty} \frac{1}{n^p} \). Knowing about p-series is crucial when analyzing other series.

Here鈥檚 the deal with p-series:
  • If \( p > 1 \), the series converges.
  • If \( p \leq 1 \), the series diverges.
For instance, the series \( \sum_{n=1}^{fty} \frac{1}{n^2} \) with \( p = 2 \) converges because 2 is greater than 1.

Why is this important? Because understanding if an unknown series resembles a p-series helps you conclude its behavior with certainty, aiding in comparison testing strategies.
Convergent Series
A convergent series is one where the sum of its infinite terms approaches a finite value. This is a core concept in calculus and analysis when dealing with infinite sums.

Understanding series convergence involves several key ideas:
  • A convergent series abides by the rule \( \lim_{n \to fty} S_n = S \), where \( S_n \) is the nth partial sum, and \( S \) is a fixed number.
  • Such a series behaves in a manner where its terms progressively and infinitely approach a ceiling, never surpassing it.
  • This concept is vital because many processes in mathematics, like integration and function approximation, hinge on determining convergence.
In practice, discerning if a series is convergent or divergent enables predicting behaviors and outcomes in mathematical problems, which is why tests 鈥� such as those discussed above 鈥� are valuable tools.

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Most popular questions from this chapter

Find the radius and interval of convergence for each series. In part \(c\) ), do not attempt to determine whether the endpoints are in the interval of convergence. (a) \(\sum_{n=0}^{\infty} n x^{n}\) (b) \(\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) (c) \(\sum_{n=1}^{\infty} \frac{n !}{n^{n}}(x-2)^{n}\) (d) \(\sum_{n=1}^{\infty} \frac{(n !)^{2}}{n^{n}}(x-2)^{n}\) (e) \(\sum_{n=1}^{\infty} \frac{(x+5)^{n}}{n(n+1)}\)

Find a power series representation for \(2 /(1-x)^{3}\).

For each function, find the Maclaurin series or Taylor series centered at \(a,\) and the radius of convergence. (a) \(\cos x\) (b) \(e^{x}\) (c) \(1 / x, a=5\) (d) \(\ln x, a=1\) (e) \(\ln x, a=2\) (f) \(1 / x^{2}, a=1\) (g) \(1 / \sqrt{1-x}\) (h) Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \(x^{3}\) term). (i) Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \(x \cos \left(x^{2}\right)\) (j) Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \(x e^{-x}\)

Explain why \(\sum_{n=1}^{\infty} \frac{5}{2^{1 / n}+14}\) diverges.

Determine whether each series converges absolutely, converges conditionally, or diverges. $$\sum_{n=2}^{\infty}(-1)^{n} \frac{1}{\ln n}$$
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