/*! 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 25 \(\lim _{n \rightarrow \infty} \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 25

\(\lim _{n \rightarrow \infty} \frac{1}{n}\left(\sin ^{2} \frac{\pi}{2 n}+\sin ^{2} \frac{2 \pi}{2 n}+\ldots . . .+\sin ^{2} \frac{n \pi}{2 n}\right)\) is equal to (1) \(\frac{2}{\pi}\) (2) \(\frac{\pi}{2}\) (3) \(\frac{1}{2}\) (4) 1

Short Answer

Expert verified
The limit is \(\frac{1}{2}\).\)

Step by step solution

01

- Understand the Limit Expression

Consider the limit expression \(\lim_{{n \to \infty}} \frac{1}{n}\bigg(\sin^2 \frac{\pi}{2n} + \sin^2 \frac{2\pi}{2n} + \ldots + \sin^2 \frac{n\pi}{2n}\bigg)\). Since the argument of the sine function is small for large n, we can use the approximation that \(\sin x \approx x for \small x.\)
02

- Apply Small Angle Approximation

For large n, using \(\sin x \approx x\), the expression becomes \[\frac{1}{n} \sum_{k=1}^{n} \left(\sin^2 \frac{k\pi}{2n}\right) \approx \frac{1}{n} \sum_{k=1}^{n} \left(\frac{k\pi}{2n}\right)^2.\]
03

- Simplify the Summation Using Approximation

Rewrite the sum \[\frac{1}{n} \sum_{k=1}^{n} \left(\frac{k\pi}{2n}\right)^2 = \frac{\pi^2}{4n^3} \sum_{k=1}^{n} k^2.\] The sum of squares formula is given by \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}.\)
04

- Insert and Evaluate the Series

Substituting the formula for the sum of squares, we have \[\frac{\pi^2}{4n^3} \cdot \frac{n(n+1)(2n+1)}{6} = \frac{\pi^2}{4n^3} \cdot \frac{2n^3 + 3n^2 + n}{6}.\]
05

- Simplify and Take the Limit

Simplify the expression: \[\frac{\pi^2}{4n^3} \cdot \frac{2n^3 + 3n^2 + n}{6} = \frac{\pi^2}{4n^3} \cdot \frac{2n^3 + O(n^2)}{6} \approx \frac{\pi^2}{4n^3} \cdot \frac{2n^3}{6} = \frac{\pi^2 \cdot 2}{24} = \frac{\pi^2}{12}.\] Taking the limit as \(n \to \infty\), we obtain \[\lim_{{n \to \infty}} \frac{\pi^2}{12} = \frac{1}{2}.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Limit Evaluation
Limit evaluation is a core concept needed for solving many mathematical problems, especially those in calculus. When you evaluate a limit, you determine the value that a function approaches as its input approaches a certain point. For the given problem, the limit of a summation involving sine squared terms as n approaches infinity must be evaluated. This involves a few critical steps:
  • Recognize that the sine function's arguments become very small for large n.
  • Use approximations to simplify the expression.
  • Apply mathematical formulas to handle summations.
By following these steps, complex expressions can be evaluated accurately, even when they involve infinite series.
Small Angle Approximation
Small angle approximation is a useful tool in trigonometry. It simplifies calculations by approximating trigonometric functions for small values of their arguments. For the sine function, the approximation \(\sin(x) \approx x\) holds true when x is close to zero. This simplification is pivotal in the given exercise, especially since \(\sin^2(x)\) can be approximated as \[ x^2 \].
This approximation allows us to transform the limit expression from one involving multiple sine terms to a simpler form involving squares of fractions.

For instance:
  • Transform \(\sin^2 \left( \frac{k \pi}{2n} \right)\) to \(\left( \frac{k \pi}{2n}\right)^2\).
This simplification significantly reduces complexity and makes it easier to apply summation formulas subsequently.
Summation Formulas
Summation formulas are crucial for solving series and understanding patterns within numbers. They allow us to sum up sequences of numbers efficiently. For the given problem, the formula for the sum of squares is particularly useful: \[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \].
This formula helps in transforming a lengthy summation into a manageable expression. Here's how it applies in the exercise:
  • Replace the sum of the series of squares in the limit expression.
  • Simplify the resulting expression to evaluate the limit as n approaches infinity.
Summation formulas thus provide a structured approach to handle and simplify complex sums, leading to an accurate evaluation of limits in problems similar to this.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Which is not correctly matched? (1) Basic strength of oxides \(\mathrm{Cs}_{2} \mathrm{O}<\mathrm{Rb}_{2} \mathrm{O}<\mathrm{K}_{2} \mathrm{O}<\mathrm{Na}_{2} \mathrm{O}<\mathrm{Li}_{2} \mathrm{O}\) (2) Stability of peroxides \(\mathrm{Na}_{2} \mathrm{O}_{2}<\mathrm{K}_{2} \mathrm{O}_{2}<\mathrm{Rb}_{2} \mathrm{O}_{2}<\mathrm{Cs}_{2} \mathrm{O}_{2}\) (3) Stability of bicarbonates \(\mathrm{LiHCO}_{3}<\mathrm{NaHCO}_{3}<\mathrm{KHCO}_{3}<\mathrm{RbHCO}_{3}<\) \(\mathrm{CsHCO}_{3}\) (4) Melting point \(\mathrm{NaF}<\mathrm{NaCl}<\mathrm{NaBr}<\mathrm{Nal}\) (1) 1 and 4 (2) 1 and 3 (3) 1 and 2 (4) 2 and 3

A spherical acetone drop evaporates at a rate proportional to its surface area at that instant. The radius of this drop initially is \(3 \mathrm{~mm}\) and after one hour it is found to be \(2 \mathrm{~mm}\). If \(\mathrm{r}(\mathrm{t})\) represents radius of the drop at time 't', Then (1) \(r(t)=3-t\) (2) \(r(t)=t-3\) (3) \(r(t)=3+t^{2}-2 t\) (4) \(r(t)=3-t^{3}\)

The equation of a simple harmonic wave is given by $$ y=3 \sin \frac{\pi}{2}(50 t-x) $$ Where \(x\) and \(y\) are in meters and \(t\) is in seconds. The ratio of maximum particle velocity to the wave velocity is (1) \(2 \pi\) (2) \(\frac{3}{2} \pi\) (3) \(3 \pi\) (4) \(\frac{2}{3} \pi\)

If refractive index and dielectric constant of the material of a glass slab is \(\frac{3}{2}\) and 3 respectively then magnetic permeability of the glass is (In \(\mathrm{Sl}\) system) (1) \(5.32 \times 10^{-7}\) (2) \(3.94 \times 10^{-7}\) (3) \(10^{-7}\) (4) \(9.42 \times 10^{-7}\)

Which of the following has been named correctly? (1) \(\mathrm{S}_{2} \mathrm{O}_{3}{ }^{2-}\) thiosulphite ion (2) \(\mathrm{N}_{3}\) - nitride ion (3) HAsO \(_{3}^{-2}\) monohydrogenarsenite ion (4) \(\mathrm{H}_{2} \mathrm{PO}_{4}\) - dihydrogenphosphite ion
See all solutions

Recommended explanations on English Textbooks

Key Concepts in Language and Linguistics

Read Explanation

Email

Read Explanation

Discourse

Read Explanation

Language and Social Groups

Read Explanation

Phonetics

Read Explanation

Essay Plan

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.