Problem 35 - Calculus: Early Transcendentals

Question

Euler also found the sum of the \( p- \) series with \( p = 4: \) \( \zeta (4) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4} = \frac {\pi^4}{90} \) Use Euler's result to find the sum of the series. (a) \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{n} \right)^4 \) (b) \( \displaystyle \sum_{k = 5}^{\infty} \frac {1}{(k - 2)^4} \)

Step-by-Step Solution

Answer

(a) \( \frac{9\pi^4}{10} \), (b) \( \frac{\pi^4}{90} - 1 - \frac{1}{16} \). \)

Step 1: Understanding the P-Series Formula

Euler's result shows that the sum of the series for \( p = 4 \) is given by \( \zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} \). This is a special harmonic series where each term is the reciprocal of \( n^4 \).

Step 2: Solving Part (a)

For the series \( \sum_{n=1}^{\infty} \left( \frac{3}{n} \right)^4 \), we need to factor out the constant \( 3^4 \) from the series: \[ \sum_{n=1}^{\infty} \left( \frac{3}{n} \right)^4 = 3^4 \sum_{n=1}^{\infty} \frac{1}{n^4} = 81 \sum_{n=1}^{\infty} \frac{1}{n^4}. \] Using Euler's result, this becomes \[ 81 \times \frac{\pi^4}{90}. \] Simplifying gives \( \frac{81\pi^4}{90} = \frac{9\pi^4}{10} \).

Step 3: Solving Part (b)

The series \( \sum_{k=5}^{\infty} \frac{1}{(k-2)^4} \) is equivalent to starting the original \( p=4 \) series from \( n=3 \). Hence, it's essentially \( \sum_{n=3}^{\infty} \frac{1}{n^4} = \zeta(4) - \left( \frac{1}{1^4} + \frac{1}{2^4} \right). \) We first compute the small terms: \( \frac{1}{1^4} = 1 \) and \( \frac{1}{2^4} = \frac{1}{16} \). The sum \( \zeta(4) \) is \( \frac{\pi^4}{90} \), so: \[ \sum_{n=3}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} - 1 - \frac{1}{16}. \] Simplify this calculation to find the precise result.

Key Concepts

Harmonic Series

The harmonic series is one of the simplest and most famous examples of a series in mathematics. It is generally expressed as:

• \( \sum_{n=1}^{\infty} \frac{1}{n} \)
• This represents the sum of the reciprocals of all positive integers.

Harmonic series are interesting because, although the terms get steadily smaller, the series as a whole actually diverges. In other words, the sum of these infinitely many diminishing terms continues to grow without bound. This makes the harmonic series a classic example when studying convergence and divergence in mathematical analysis. However, when we start dealing with series like \( p- \)series, where each term is \( \frac{1}{n^p} \), things change depending on the value of \( p \). For any \( p > 1 \), these series actually converge, a fascinating contrast to the harmonic series itself.

Convergence

Convergence refers to the behavior of an infinite series to approach a fixed value. When we say a series converges, it means as we keep summing the terms indefinitely, they tend to approach a specific number.

• If \( p > 1 \) in a \( p- \)series, the series is known to converge.
• Conversely, if \( p \le 1 \), the series will diverge or spread without reaching any finite limit.

A key question in calculus and analysis is determining whether a given series actually converges and, if so, finding out what it converges to. This is vital in understanding many physical and theoretical concepts where accurately summing an infinite series is crucial.

Zeta Function

The zeta function is a mathematical construct that extends the notion of a \( p- \)series and plays a huge role in number theory. It is commonly written as \( \zeta(p) \) and defined by the formula:

• \( \zeta(p) = \sum_{n=1}^{\infty} \frac{1}{n^p} \)
• For \( p > 1 \), this function converges and is well defined.

Especially famous is the Riemann zeta function, which deals with understanding the distribution of prime numbers. The significance of the zeta function goes beyond mere summing; it impacts different fields from quantum physics to cryptography. This function, like in the example with \( p = 4 \) leading to Euler's specific result, highlights the deep and interconnected nature of mathematics.

Euler's Result

Euler made a groundbreaking discovery by calculating the sum of the zeta function for \( p = 4 \). His result is:

• \( \zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} \)
• This numerical result connects the sum of the series to the transcendental number \( \pi \).

Euler's work with these sums provided a foundation for later developments in analysis and number theory, such as the Riemann hypothesis. Using Euler's result, solving complex series becomes more feasible. For instance, problems like calculating sums with adjusted terms by factoring constants or starting from a different index are simplified. Euler's exploration and results offer powerful insights into the nature of numbers and series convergence.