Series

Let's evaluate the integral .

We can use the property of definite integrals for simplification. Consider the substitution . Then , or .

Changing the limits of integration accordingly, when , and when , .

The integral becomes:

This can be rewritten using the given symmetrical property:

Using symmetry, it's known that:

For our integral:

Notice that due to the symmetrical property, because changing variables from gives the same integral:

Therefore, , which results in:

The value of the original integral is:

Thus, the correct answer is .

To find the sum to infinity of the series , let's analyze the pattern in the terms.

The given series is not a standard geometric series. However, we can express the term in a generalized form for recognizing any underlying pattern.

The general term of the series can be written as: , where

Thus, the terms of the series can be expressed as:

• The first term:
• The second term:
• The third term:
• The fourth term:

The nth term is given by:

To find the sum to infinity, observe or calculate the initial terms for convergence:

• As , both the numerator grows linearly, and the denominator grows exponentially.
• Series is converging as each term is reduced in size exponentially.

The sum of the series is computed using Cauchy-product or detecting patterns via known convergence of non-standard series:

The correct answer is: 3