Chapter 1: Q5P (page 22)

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case $\sum_{n = 1}^{\infty} \frac{x^{n}}{{(n!)}^{2}}$

Short Answer

Expert verified

All power series convergence intervals are X values.

Step by step solution

Significance of ratio test

The ratio test as follows:

Solution steps:

First,find $ρ_{n} = | \frac{a_{n + 1}}{a_{n}} | .$

Second, $ρ = \lim_{n \to \infty} p_{n} .$ .

Third, $p < 1$ t he series converges, if $p > 1$ the series diverges.

Finding convergence

Let us consider the power series

$\sum_{n = 1}^{\infty} \frac{x^{n}}{{(n!)}^{2}}$

And we would like to find the interval of convergence of the power series.

We will be using the ratio test, as follows:

Solution steps:

First,find role="math" localid="1658820727600" $ρ_{n} = | \frac{a_{n + 1}}{a_{n}} |$ .

Second, $ρ = \lim_{n \to \infty} p_{n} .$

Third, $p < 1$ the series converges, if $p > 1$ the series diverges.

role="math" localid="1658820961551" $ρ_{n} = | \frac{a_{n + 1}}{a_{n}} | = |\frac{\frac{x^{n + 1}}{((n + 1)!)^{2}}}{\frac{x^{n}}{(n!)^{2}}}| ρ_{n} = |\frac{(x^{n + 1} (n!)^{2}}{((n + 1)!)^{2} x^{n}}|$

Finding the points of divergence

Notice that $(n + 1)! = (n + 1) n!$ . Hence

$ρ_{n} = |\frac{x}{(n + 1)^{2}}| ρ = \lim_{n \to \infty} |\frac{x}{(n + 1)^{2}}| = 0$

$p < 1$ For all the values of x .

Concluding statement

All power series convergence intervals are x values.

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Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case $\sum_{n = 1}^{\infty} \frac{x^{n}}{{(n!)}^{2}}$

Short Answer

Step by step solution

Significance of ratio test

Finding convergence

Finding the points of divergence

Concluding statement

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Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case∑n=1∞xn(n!)2

Short Answer

Step by step solution

Significance of ratio test

Finding convergence

Finding the points of divergence

Concluding statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case $\sum_{n = 1}^{\infty} \frac{x^{n}}{{(n!)}^{2}}$