Question

For each of the given series, make a change of summation index so that the new sum contains only nonzero terms. Replace constants expressed in terms of trigonometric functions by equivalent numerical values [for example, \(\cos n \pi=(-1)^{n}\) ]. $$\sum_{n=1}^{\infty} \frac{4 \cos (n \pi / 2)}{n^{2} \pi^{2}+1} \cos (n \pi x)$$

Short answer

Question: Determine the equivalent series with only nonzero terms for the given series: $$\sum_{n=1}^{\infty} \frac{4 \cos(n\pi / 2)}{(n^{2} \pi^{2}+1)} \cos(n \pi x)$$ Answer: The equivalent series with only nonzero terms is: $$\sum_{m=1}^{\infty} \frac{4 (-1)^m}{((2m - 1)^{2} \pi^{2}+1)} \cos((2m - 1) \pi x)$$

Step-by-step solution

Identify nonzero terms in the cosine function

To find the nonzero terms in the series, we need to determine the values of n for which the cosine function is nonzero. The cosine function \(\cos(n\pi / 2)\) is zero when its argument is a multiple of \(\pi\) (i.e., \(n\pi / 2 = k\pi\) for some integer k). To make the cosine function nonzero, we need to exclude these values of n.

Make a change of summation index

Considering the analysis above, let's change the summation index to allow only odd values of n, as only odd values of n will result in nonzero terms for \(\cos(n\pi / 2)\). We can use a new index m such that: $$n = 2m - 1$$ where m is an integer starting from 1. Now, we need to rewrite the series in terms of m.

Rewrite the series in terms of the new index

Using the new index and the properties of the cosine function, let's rewrite the given series in terms of m: $$\sum_{m=1}^{\infty} \frac{4 \cos ((2m - 1)\pi / 2)}{((2m - 1)^{2} \pi^{2}+1)} \cos((2m - 1) \pi x)$$

Replace trigonometric constants with numerical values

Using the identity \(\cos(n\pi) = (-1)^n\), we can replace the cosine function in the numerator with its numerical equivalent: $$\sum_{m=1}^{\infty} \frac{4 (-1)^m}{((2m - 1)^{2} \pi^{2}+1)} \cos((2m - 1) \pi x)$$ Now, we have found the equivalent series with only nonzero terms: $$\sum_{m=1}^{\infty} \frac{4 (-1)^m}{((2m - 1)^{2} \pi^{2}+1)} \cos((2m - 1) \pi x)$$

Key concepts

Summation Index Transformation

In mathematics, we often work with series that converge under certain constraints. Sometimes, it might be beneficial to change the summation index to exclude terms that do not contribute to the sum. This is known as summation index transformation.
When we change the summation index, we effectively adjust how we count each term in the series. This can be crucial if the original index causes some terms to become zero, making them irrelevant to the final summation.
A common practice is to substitute the index with a formula that limits its values to a specific subset. For example, if we want to focus only on odd values from a series, we can let \( n = 2m - 1 \) where \( m \) is an integer. This transformation avoids even-numbered terms, which could be zero in a trigonometric function, thus optimizing the series for nonzero elements.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a vital role in analyzing periodic phenomena in math and physics. These functions can sometimes become zero or take on numerical values depending on their input arguments, which can further influence series evaluations.
For example, the function \( \cos(n\pi/2) \) becomes zero whenever its argument is a multiple of \( \pi \). This behavior can set some terms in a series to zero inadvertently.
By altering the summation index or substituting trigonometric identities, such as \( \cos(n\pi) = (-1)^n \), it's possible to convert trigonometric constants in series into numerical values. This step simplifies series computations by replacing the cosine terms with appropriate numbers, facilitating easier analysis and manipulation.

Series Convergence

Understanding series convergence is crucial for ensuring that the infinite series sums to a finite number. A series is said to converge if its partial sums approach a specific finite value as the number of terms increases indefinitely.
To assess convergence, one might apply various tests such as the ratio test, root test, or comparison test. Convergence is especially important when dealing with infinite series in calculus and analysis because it assures us that the series has a meaningful sum.
In our exercise about altering a summation index, we change terms to support convergence by focusing on nonzero elements, which helps maintain the series converging appropriately. By excluding zero terms and using transformations, we lay down a structure that promotes convergence, ensuring that all the non-zero elements contribute to the finite sum of the series.