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, \(\cos n \pi=(-1)^{n}\) ]. $$\sum_{n=1}^{\infty} \frac{-1+(-1)^{n}}{n^{2} \pi^{2}} \cos (n \pi x)$$

Short answer

Answer: The simplified sum of the given series is 0.

Step-by-step solution

Make a change of summation index

To achieve this, we need to notice that \((-1)^{n}\) yields 0 only for even values of n. Therefore, we can make a change of summation index by setting \(m=2n\) for even values of n. The summation will now go from 1 to infinity over the new index m. The given series can be expressed as follows: $$\sum_{m=1}^{\infty} \frac{-1+(-1)^{2m}}{(2m)^{2} \pi^{2}} \cos (2m \pi x)$$

Evaluate the trigonometric function

We can simplify this expression by evaluating the trigonometric function and the expression in the numerator as follows: $$(-1)^{2m}=1 \text{ and } \cos(2m\pi x) = \cos(2\pi mx)$$ Taking these into account, we can rewrite the expression as: $$\sum_{m=1}^{\infty} \frac{(-1)+1}{(2m)^{2} \pi^{2}} \cos (2 \pi m x)$$

Simplify the series

Now, we can simplify the expression further: $$\sum_{m=1}^{\infty} \frac{0}{(2m)^{2} \pi^{2}} \cos (2 \pi m x)$$

Conclusion

Since the series converges to 0, the sum containing only non-zero terms can be expressed as: $$\sum_{m=1}^{\infty} 0$$

Key concepts

Summation Index Change

The concept of changing the summation index is an important mathematical technique used to simplify series and make expressions more manageable. In essence, it involves redefining the terms of a series to start from a different index while maintaining the overall integrity of the summation. Often, this is done to eliminate zero or unnecessary terms. When we encounter a series like
\[ \sum_{n=1}^{\infty} \frac{(-1)+(-1)^{n}}{n^{2} \pi^{2}} \cos (n \pi x) \]
we can observe that the terms produce zero at certain intervals, which are redundant to the sum. By setting an index change such as
\( m = 2n \),
we bypass the zero values and reformulate the series with non-zero elements only, making the overall calculation more efficient.

Convergence of Series

Understanding the convergence of series is essential for mathematical analysis, particularly in the context of infinite series. A series converges if the sum of its terms approaches a finite value as the number of terms grows to infinity. Conversely, if the sum grows without bound or oscillates indefinitely, the series diverges.

In the given example, after the simplification process, we are left with a series in the form:
\[ \sum_{m=1}^{\infty} 0 \]
Which evidently has a sum of zero for all terms. This simplification indicates the convergence of the original series, as adding zero quantities indefinitely still results in zero—a finite and concrete value.

Trigonometric Functions in Series

Trigonometric functions often appear in the study of series, especially when dealing with waveforms or periodic phenomena. These functions, like cosine and sine, possess properties that can drastically simplify evaluations when used in series. For instance, cosine and sine functions are periodic, meaning they repeat their values in regular intervals, and they have specific symmetries that can be utilized during simplification.

During the simplification process, we utilized the fact that cosine is an even function, which means
\( \cos(-\theta) = \cos(\theta) \).
Additionally, pairing trigonometric functions with powers of \( (-1)^n \) can often lead to significant and useful simplifications, as seen when addressing the term
\( (-1)^{2m} = 1 \).
These properties can reduce many terms in the series to zero, simplifying the sum tremendously.

Fourier Series

The Fourier series is an advanced mathematical tool used to decompose periodic functions into a sum of sine and cosine functions multiplied by coefficient constants. This decomposition allows us to analyze and understand various functions through the lens of simple trigonometric series.

Although the original exercise didn't directly deal with a Fourier series, the form of the given series and the process of transforming it through the principles of orthogonality and index changes bears resemblance to methods used to manipulate Fourier series. In the context of a Fourier series, the cosine components with a zero-valued coefficient would effectively contribute nothing to the function's approximation, similar to how terms were eliminated in our simplified exercise.