Chapter 1: Q17P (page 1)

Use the integral test to show that $\sum_{n = 0}^{\infty} e^{- n^{2}}$ converges.

Short Answer

Expert verified

The series $\sum_{n = 0}^{\infty} e^{- n^{2}}$ is convergent.

Step by step solution

Definition of Gaussian integral, convergent, and divergent

The Gaussian integral which is also known as Poisson integral. The area of Gaussian function ${f(x)=e}^{- x^{2}}$ and over the x-axis is as follows:

$\int_{- \infty}^{\infty} e^{- x^{2}} dx = \sqrt{π}$

Then, half of the area of Gaussian function and -axis is: $\int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2}$ .

If the partial sum localid="1664193252645" $S_{n}$ of an infinite series tend to a limit Sthen the series is called convergent. If the partial sums $S_{n}$ of an infinite series do not approach a limit, the series is called divergent. The limiting value S is called the sum of the series.

Apply the integral test

Given that the limit is $\sum_{n = 0}^{\infty} e^{- n^{2}}$ .

Now, apply the integral test as:

$\int_{0}^{\infty} e^{- n^{2}} d n$

This is half of the area of Gaussian function and x-axis which is, $\int_{0}^{\infty} e^{- n^{2}} d n = \frac{\sqrt{π}}{2}$ .

The integral value is finite, therefore the given series $\sum_{n = 0}^{\infty} e^{- n^{2}}$ converges.

Solve the integral

Now, compare with the integral $\int_{}^{\infty} e^{- n} d n$ .

Evaluate the integral is as follows:

$\int_{}^{\infty} e^{- n} d n = {[\frac{e^{- n}}{- 1}]}^{\infty} = - [e^{- \infty}] = 0$

The integral value is finite, therefore the integral $\int_{}^{\infty} e^{- n} d n$ converges. Here, both the integral converges.

Hence, it is proved that series $\sum_{n = 0}^{\infty} e^{- n^{2}}$ converges.

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Use the integral test to show that $\sum_{n = 0}^{\infty} e^{- n^{2}}$ converges.

Short Answer

Step by step solution

Definition of Gaussian integral, convergent, and divergent

Apply the integral test

Solve the integral

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Use the integral test to show that∑n=0∞e-n2 converges.

Short Answer

Step by step solution

Definition of Gaussian integral, convergent, and divergent

Apply the integral test

Solve the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

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Study anywhere. Anytime. Across all devices.

Use the integral test to show that $\sum_{n = 0}^{\infty} e^{- n^{2}}$ converges.