Chapter 1: Q6MP (page 1)

Test for convergence: $\sum_{n = 2}^{\infty} \frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1}$

Short Answer

Expert verified

The limit is finite and the series converges.

Step by step solution

Concept used to prove that the series converges

Let $0 ⩽ a_{n} ⩽ b_{n}$ for all $n$ .

Ifrole="math" localid="1664276577803" $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ is also converges.

If $\sum_{n = 1}^{\infty} a_{n}$ diverges, then $\sum_{n = 1}^{\infty} b_{n}$ is also diverges.

Calculation to show that the series converges

Let the series $a_{n} = \sum_{n = 2}^{\infty} \frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1}$ .

Consider the series $\sum_{n = 2}^{\infty} \frac{\sqrt{n}}{n^{2}} = \sum_{n = 2}^{\infty} \frac{1}{n^{\frac{3}{2}}}$ .

Let $b_{n} = \frac{1}{n^{\frac{3}{2}}}$ .

The series $\sum b_{n} = \sum \frac{1}{n^{\frac{3}{2}}}$ is convergent series by integral test.

Take $l i m_{n \to \infty} \frac{a_{n}}{b_{n}}$ .

Simplify the above expression.

$l i m_{n \to \infty} \frac{a_{n}}{b_{n}} = l i m_{n \to \infty} (\frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1} \div \frac{1}{n^{\frac{3}{2}}}) = l i m_{n \to \infty} (\frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1} \cdot n^{\frac{3}{2}}) = l i m_{n \to \infty} (\frac{\sqrt{n} \sqrt{1 - \frac{1}{n}}}{n^{2} ({(1 + \frac{1}{n})}^{2} - \frac{1}{n^{2}})} \cdot n^{\frac{3}{2}})$

Solve the limit.

width="244">limn→∞anbn=1-1w1+1w2-1(s)2

Hence, the limit is finite and the series converges.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Test for convergence: $\sum_{n = 2}^{\infty} \frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1}$

Short Answer

Step by step solution

Concept used to prove that the series converges

Calculation to show that the series converges

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Electricity

Further Mechanics and Thermal Physics

Radiation

Measurements

Quantum Physics

Study anywhere. Anytime. Across all devices.

Test for convergence:∑n=2∞n-1(n+1)2-1

Short Answer

Step by step solution

Concept used to prove that the series converges

Calculation to show that the series converges

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Electricity

Further Mechanics and Thermal Physics

Radiation

Measurements

Quantum Physics

Study anywhere. Anytime. Across all devices.

Test for convergence: $\sum_{n = 2}^{\infty} \frac{\sqrt{n - 1}}{{(n + 1)}^{2} - 1}$