Chapter 1: Q20P (page 41)

By computer or tables, find the exact sum of each of the following series.
$(a) \sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} (b) \sum_{n = 1}^{\infty} \frac{n^{3}}{n!} (c) \sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}}$

Short Answer

Expert verified

The sum of the series $\sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} = \frac{1}{8}$ .
The sum of the series $\sum_{n = 1}^{\infty} \frac{n^{3}}{n!} = 13.5914091422 = 5 e$ .
The sum of the series $\sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}} = \frac{9}{4}$ .

Step by step solution

Given Information

The given series are, i.e.,

$(a) \sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} (b) \sum_{n = 1}^{\infty} \frac{n^{3}}{n!} (c) \sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}}$

Definition of Sum of a Series

The total of all the terms in a sequence can be solved as the sum of a series.

(a) Find the exact sum of the series ∑n=1∞n(4n2-1)2

Use the MATLAB tool to calculate the exact sum of the series,

Hence, the sum of the series $\sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} = \frac{1}{8}$ .

(b) Find the exact sum of the series ∑n=1∞n3n!

Use the MATLAB tool to calculate the exact sum of the series,

Hence, the sum of the series $\sum_{n = 1}^{\infty} \frac{n^{3}}{n!} = 13.5914091422 = 5 e$ .

(c) Find the exact sum of the series ∑n=1∞n(n+1)3n

Use the MATLAB tool to calculate the exact sum of the series $\sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}}$ ,

Hence, the sum of the series $\sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}} = \frac{9}{4}$ .

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.

By computer or tables, find the exact sum of each of the following series.
$(a) \sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} (b) \sum_{n = 1}^{\infty} \frac{n^{3}}{n!} (c) \sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}}$

Short Answer

Step by step solution

Given Information

Definition of Sum of a Series

(a) Find the exact sum of the series ∑n=1∞n(4n2-1)2

(b) Find the exact sum of the series ∑n=1∞n3n!

(c) Find the exact sum of the series ∑n=1∞n(n+1)3n

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Torque and Rotational Motion

Kinematics Physics

Scientific Method Physics

Work Energy and Power

Dynamics

Study anywhere. Anytime. Across all devices.

By computer or tables, find the exact sum of each of the following series.(a)∑n=1∞n(4n2-1)2(b)∑n=1∞n3n!(c)∑n=1∞n(n+1)3n

Short Answer

Step by step solution

Given Information

Definition of Sum of a Series

(a) Find the exact sum of the series ∑n=1∞n(4n2-1)2

(b) Find the exact sum of the series ∑n=1∞n3n!

(c) Find the exact sum of the series ∑n=1∞n(n+1)3n

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Torque and Rotational Motion

Kinematics Physics

Scientific Method Physics

Work Energy and Power

Dynamics

Study anywhere. Anytime. Across all devices.

By computer or tables, find the exact sum of each of the following series.
$(a) \sum_{n = 1}^{\infty} \frac{n}{{(4 n^{2} - 1)}^{2}} (b) \sum_{n = 1}^{\infty} \frac{n^{3}}{n!} (c) \sum_{n = 1}^{\infty} \frac{n (n + 1)}{3^{n}}$