Question

Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90},\) show that \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}}=\frac{7 \pi^{4}}{720}.\) (Assume the result of Exercise 63.)

Short answer

Answer: \(\frac{7 \pi^{4}}{720}\)

Step-by-step solution

Write down the given series and the desired series

We are given the series: $$\sum_{k=1}^{\infty} \frac{1}{k^{4}} = \frac{\pi^{4}}{90}$$ And we aim to find: $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} = S$$

Add the given series to the desired series

In order to manipulate the desired series, add the given series to it: $$\sum_{k=1}^{\infty} \frac{1}{k^{4}} + \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} = \frac{\pi^{4}}{90} + S$$ $$(2) \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} = \frac{\pi^{4}}{90} + S$$

Rearrange the equation to find the desired series

Now, solve the equation for \(S\), which represents the desired series: $$S = \frac{(2) \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} - \frac{\pi^{4}}{90}}{3} = \frac{\pi^{4}}{180} - \frac{\pi^{4}}{90} = \frac{\pi^{4}}{60} - \frac{\pi^{4}}{30} = \frac{7 \pi^{4}}{720}$$ Thus, we have shown that: $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} = \frac{7 \pi^{4}}{720}$$

Key concepts

Riemann Zeta Function

The Riemann Zeta Function, denoted by \( \zeta(s) \), is a fundamental tool in number theory and complex analysis, defined for complex numbers \( s \) with a real part greater than 1. This function is an infinite series represented as:

• \( \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} \)

Here, each term contributes the reciprocal of \( k \) raised to the power of \( s \). For special values, such as \( s = 2, 4, 6, \ldots \), the series converges to well-known constants, including expressions involving \( \pi \), like \( \zeta(4) = \frac{\pi^4}{90} \).
The Riemann Zeta Function extends beyond real numbers and is central to the distribution of prime numbers. For values where the real part of \( s \) is less than or equal to 1, it is analytically continued, except at \( s = 1 \), where it diverges. When we deal with alternating series, like in the given problem, the zeta function can be adjusted by transformation principles that account for alternating signs.

Convergence of Series

Convergence of an infinite series is a crucial concept in analysis. For a series to converge, the sum of its infinite terms must approach a specific limit. This entails the partial sums of the series getting closer to a finite value as more terms are added.
Consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^{4}} \) from our example. This series is known to converge, and its sum is given as \( \frac{\pi^4}{90} \). Its convergence is assured since the terms \( \frac{1}{k^4} \) become very small rapidly due to the fourth power in the denominator. The convergence is guaranteed when the exponent in the series \( \frac{1}{k^p} \) is greater than 1, known as the p-series test.

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

In alternating series like \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4}} \), Liebnitz's test is commonly used to ensure convergence. The conditions include that the terms' magnitude decreases and approaches zero as \( k \) increases.

Infinite Series

Infinite series can be seen in mathematics where a sequence extends indefinitely. Understanding this concept involves evaluating the behavior and properties of summing an endless number of terms. For example, in our exercise, both series involved have an infinite number of terms, each term given by \( \frac{1}{k^{4}} \), but with different signs.
Infinite series can be:

• Convergent - where they sum to a finite limit, like our harmonic series in the example.
• Divergent - where no finite limit is approached as more terms are added.

Exploring the realm of infinite series aids in understanding various mathematical and physical phenomena, including geometric series, power series, and even Fourier series in signal processing. In applied mathematics, series are useful for approximating functions and solving complex problems that cannot be addressed by finite processes. In our problem, by considering an infinite alternating series, it allows us to calculate the adjusted sum using the known values of the regular infinite series.