Question

Assume that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\) (Exercises 65 and 66 ) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.

Short answer

Answer: The sum of the reciprocals of the squares of odd positive integers is equal to \(\frac{\pi^2}{8}\).

Step-by-step solution

Write down the sum of the reciprocals of the squares of all positive integers

We are given the infinite sum \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\).

Separate the terms for odd and even integers

To find the sum of the reciprocals of the squares of odd positive integers, we need to split the sum into two parts: one for odd integers and one for even integers. We can do this using the property of even and odd integers: odd integers can be written as \(2n-1\) and even integers as \(2n\) where n is any positive integer. So, the sum for odd integers is \(\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}}\) and the sum for even integers is \(\sum_{n=1}^{\infty} \frac{1}{(2n)^{2}}\).

Rewrite the given sum as sum of these two sums

Using the separated sums for odd and even integers, rewrite the given sum as: \(\frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}} + \sum_{n=1}^{\infty} \frac{1}{(2n)^{2}}\).

Find the sum of reciprocals of squares of even integers

In order to simplify the sum for even integers, take out a factor of 4 from the denominator, since \((2n)^2 = 4n^2\): \(\sum_{n=1}^{\infty} \frac{1}{(2n)^{2}} = \sum_{n=1}^{\infty} \frac{1}{4n^{2}} = \frac{1}{4} \sum_{n=1}^{\infty} \frac{1}{n^{2}}\).

Rewrite our sum for odd integers using the known total sum and sum of even integer squares

Now, subtract the sum of reciprocals of squares of even integers from the total sum to find the sum for odd integers: \(\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}} = \frac{\pi^2}{6} - \frac{1}{4} \sum_{n=1}^{\infty} \frac{1}{n^{2}}\).

Calculate the final sum

Replace the sum in the equation with the known value \(\frac{\pi^2}{6}\): \(\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}} = \frac{\pi^2}{6} - \frac{1}{4} \cdot \frac{\pi^2}{6}\). Now, calculate the final sum: \(\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}} = \frac{\pi^2}{6} (1 - \frac{1}{4}) = \frac{3}{4} \cdot \frac{\pi^2}{6} = \frac{\pi^2}{8}\). Thus, the sum of the reciprocals of the squares of odd positive integers is equal to \(\frac{\pi^2}{8}\).

Key concepts

Reciprocals of Squares

When addressing the concept of reciprocals of squares, we talk about taking each positive integer, squaring it, and then taking the reciprocal of that square. Mathematically, for a positive integer k, the reciprocal of its square is \( \frac{1}{k^2} \). If you consider the sum of these reciprocals for all positive integers (1, 2, 3, ... and so on), you’re diving into an interesting area of infinite series.

The fascinating aspect of this series is that even though it contains an infinite number of terms, it is known to converge to a specific value, \( \frac{\pi^2}{6} \), as discovered by the renowned mathematician Leonhard Euler. To make this concept more approachable, it's like saying: if you could add up all these tiny fractions forever, surprisingly, they all add up to a bit over 1.64, which is an amazing testament to the peculiar beauty of infinity in mathematics.

Odd and Even Integers

The dance between odd and even integers is a fundamental rhythm in mathematics. Even integers are like partners who always come in pairs (\(2n\)), while odd integers dance to their own single beat (\(2n-1\)). Why does this matter? When looking at infinite series involving the squares of integers, we can separate the squad into two groups: one with all the odd numbers squaring off, and the other with the evens pairing up.

The way we'd express this is by writing two separate sums: one for the squares of the odd integers (\( \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}} \) ) and another for the squares of the even integers (\( \sum_{n=1}^{\infty} \frac{1}{(2n)^{2}} \) ). The technique of dividing and conquering in this manner allows mathematicians to explore intriguing properties of these infinite series, opening pathways to understanding their behaviors separately and then piecing it all together for a grand result.

Infinite Sum Convergence

When students first encounter the concept of infinite sum convergence, there's often a mix of intrigue and skepticism – how can adding up an endless list of numbers result in a finite value? Yet this is precisely what happens with certain types of infinite series. Convergence is like a lighthouse beacon: it assures us that there's solid ground ahead despite the seemingly endless sea of terms.

The sum of the reciprocals of the squares is a classic example – mathematicians have proven that as you keep adding more and more terms, the sum gets closer and closer to \( \frac{\pi^2}{6} \) but never exceeds it. Think of it as filling a glass with water drop by drop; eventually, the glass will be full, and no more water can fit in, even if you keep adding drops. That 'fullness' is the \( \frac{\pi^2}{6} \) in our series, a point where adding infinitesimally small amounts will no longer increase the total. This behavior is what mathematicians call convergence, and it's a cornerstone concept for working with infinite series.

Convergence of Series

The convergence of series is a broader concept, like the umbrella that shelters various types of series, guiding us to discern whether or not the infinite sum has a finite destination. In the world of mathematics, not all infinite series are friendly enough to settle down to a single value, but those that do, like the sum of the reciprocals of the squares of integers, hold a special badge of honor.

To determine whether a series converges, mathematicians run some tests. One could compare it to the series with the known behavior, look for patterns, or apply rigorous criteria developed over centuries. For many series, convergence isn't just about whether they add up to a specific number, but about the behavior of their terms as they progress to infinity. When teaching convergence, it's crucial to illustrate not just that a series adds up to a number, but to imbue students with an understanding of the 'why' and the 'how' — the fascinating logic behind the steady march of numbers towards their infinite horizon.