Chapter 1: Q21P (page 41)

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

Short Answer

Expert verified

The sum of the series $\sum_{n = 1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}} = 0.39711677$ .
The sum of the series $\sum_{n = 2}^{\infty} \frac{In n}{n^{2}} = 0.93754825431$ .
The sum of the series $\sum_{n = 1}^{\infty} \frac{1}{n^{n}} = 1.29128599 .$ .

Step by step solution

Given Information

The given series are, i.e.,

$(a) \sum_{n = 1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}} (b) \sum_{n = 2}^{\infty} \frac{In n}{n^{2}} (c) \sum_{n = 1}^{\infty} \frac{1}{n^{n}}$

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(n2-1)2

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

Hence, the sum of the series $\sum_{n = 1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}} = 0.39711677$ .

(b) Find the exact sum of the series ∑n=2∞In nn2

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

Hence, the sum of the series $\sum_{n = 2}^{\infty} \frac{In n}{n^{2}} = 0.93754825431$ .

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

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

Hence, the sum of the series $\sum_{n = 1}^{\infty} \frac{1}{n^{n}} = 1.29128599$ .

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

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(n2-1)2

(b) Find the exact sum of the series ∑n=2∞In nn2

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

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By computer or tables, find the exact sum of each of the following series.(a)∑n=1∞n(n2+1)2(b)∑n=2∞Innn2(c)∑n=1∞1nn

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(n2-1)2

(b) Find the exact sum of the series ∑n=2∞In nn2

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Engineering Physics

Nuclear Physics

Wave Optics

Conservation of Energy and Momentum

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

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