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Chapter 9: Problem 71

Determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \arctan n\)

Short Answer

Expert verified
Hence, according to the divergence test, this series is divergent as the limit of the sequence as n approaches infinity is not zero.

Step by step solution

01

Identify the type of series

The given mathematical expression is an infinite series where the sequence is \(arctan(n)\). So, it falls under the category of infinite series.
02

Test for divergence

Before applying any other test, a common first step is to apply the Test for Divergence. This test states that if the limit of the sequence as n approaches infinity is not zero then the series is divergent.
03

Apply the Test for Divergence

Applying the Test for Divergence by calculating the limit as n approaches infinity for \(arctan n\), it's clear that this limit is \(\frac{\pi}{2}\) which is non-zero.

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Key Concepts

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

Test for Divergence
The Test for Divergence is a fundamental tool used in determining the convergence or divergence of an infinite series. It is often the first test applied when analyzing a series. The concept is simple and straightforward. If the limit of the series' terms as they approach infinity is not zero, then the series definitely diverges. Think of it like checking if the terms shrink away to nothing.
To elaborate:
  • We calculate the limit of the sequence that makes up the series, as n (the index) tends towards infinity.
  • If this limit turns out to be anything other than zero, such as a positive number, negative number, or infinity, it's a clear sign that the series does not converge.
Remember, a non-zero limit means the terms do not disappear, so they accumulate, pushing the series towards divergence.
Infinite Series
An infinite series is a sum of an infinite sequence of numbers. It's like adding up numbers without end. When we talk about series, we talk in terms of convergence and divergence.
The structure is usually represented by a sum notation such as \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) represents the sequence of term values.
Convergence and divergence of series:
  • A series converges if the sum approaches a specific finite number as more and more terms are added.
  • If the sum keeps growing without settling towards any number, the series diverges.
This distinction is vital for understanding mathematical models and calculations that involve infinite processes.
Arc Tangent Function
The arc tangent function, written as \(\arctan(x)\), is a mathematical function used to find the angle whose tangent is \(x\). It's the inverse of the tangent function and is part of the family of trigonometric inverse functions.
The arc tangent function:
  • Returns values between -\(\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
  • Is continuous and smooth across its domain.
  • Behaves such that as \(x\) approaches infinity, \(\arctan(x)\) approaches \(\frac{\pi}{2}\).
This function is used in calculus and trigonometry, often appearing in series as seen in our exercise. It showcases how functions impact the convergence or divergence of a series.

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Most popular questions from this chapter

Prove that \(\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0\) for any real \(x\).

Evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}\) where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}k \\ 0\end{array}\right)=1\) \(\left(\begin{array}{c}0.5 \\ 4\end{array}\right)\)

show that the function represented by the power series is a solution of the differential equation. $$ y=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !}, \quad y^{\prime \prime}-y=0 $$

(a) Show that \(\int_{1}^{n} \ln x d x<\ln (n !)\) for \(n \geq 2\). (b) Draw a graph similar to the one above that shows \(\ln (n !)<\int_{1}^{n+1} \ln x d x\) (c) Use the results of parts (a) and (b) to show that \(\frac{n^{n}}{e^{n-1}}1 .\) (d) Use the Squeeze Theorem for Sequences and the result of part (c) to show that \(\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n !}}{n}=\frac{1}{e^{*}}\) (e) Test the result of part (d) for \(n=20,50\), and 100 .

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1}, \quad-1 \leq x \leq 1 $$
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