Chapter 1: Q6P (page 9)

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.
6. $\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$

Short Answer

Expert verified

The series requires further testing.

Step by step solution

Use the preliminary test

Consider the series:

$\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$

Use the preliminary test to see if this series diverges or converges, where $a_{n} = \frac{n!}{(n + 1)!}$ .

So, $\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{n!}{(n + 1)!}$ .

Check the parameter

It is known that $(n + 1)! = (n + 1) . n!$ . Hence,

$\begin{array}{rcl} \lim_{n \to \infty} a_{n} & = & \lim_{n \to \infty} \frac{n!}{(n + 1)!} \\ = & \lim_{n \to \infty} \frac{1}{(n + 1)} \\ = & \frac{1}{\infty} \\ = & 0 \end{array}$

Because $\lim_{n \to \infty} a_{n} = 0$ , this means that preliminary testing is not enough and the series will need to be further tested using other methods.

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Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.
6. $\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$

Short Answer

Step by step solution

Use the preliminary test

Check the parameter

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Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.6.∑n=1∞n!(n+1)!

Short Answer

Step by step solution

Use the preliminary test

Check the parameter

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Magnetism and Electromagnetic Induction

Turning Points in Physics

Classical Mechanics

Electrostatics

Electromagnetism

Study anywhere. Anytime. Across all devices.

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.
6. $\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$