Question

\(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$\cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

Short answer

Question: Find the limit of the sum of 1/k^2 as n approaches infinity. Answer: Based on the analysis and solution provided, the limit of the sum of 1/k^2 as n approaches infinity is equal to π^2/6.

Step-by-step solution

1. Prove the inequality involving cotangent and sum

First, we need to show the inequality: $$\sum_{k=1}^{n} \cot ^{2} k \theta<\frac{1}{\theta^{2}} \sum_{k=1}^{n} \frac{1}{k^{2}}<n+\sum_{k=1}^{n} \cot ^{2} k \theta$$ The given inequality states that: $$\cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x \quad \text { for } 0<x<\frac{\pi}{2}$$ We multiply the inequality by k^2 and sum over k from 1 to n: $$\sum_{k=1}^{n} k^{2}\cot^{2}{(k\theta)} < \sum_{k=1}^{n} k^{2} \cdot \frac{1}{k^{2}} < \sum_{k=1}^{n} k^2(1 + \cot^{2}{(k\theta)})$$ Dividing by k^2 and noting that cotangent squared is always positive, we get the inequality we're asked to prove.

2. Use the identity involving the sum of cotangents

The identity provided is: $$\sum_{k=1}^{n} \cot ^{2} k \theta=\frac{n(2 n-1)}{3},$$ for \(\theta=\frac{\pi}{2 n+1}\). Using this identity, let's rewrite the inequality from Step 1: $$\frac{n(2 n-1)}{3}<\frac{1}{\theta^{2}}\sum_{k=1}^{n} \frac{1}{k^{2}}<n+\frac{n(2 n-1)}{3}$$

3. Show the inequality involving the sum of 1/k^2 and fractions

Now, let's work on the inequality: $$\frac{n(2 n-1) \pi^{2}}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{n(2 n+2) \pi^{2}}{3(2 n+1)^{2}}$$ Since \(\theta = \frac{\pi}{2n+1}\), we get: $$\frac{1}{\theta^2} = \frac{(2n+1)^2}{\pi^2}$$ Substitute this into the inequality from Step 2: $$\frac{n(2 n-1) \pi^{2}}{3(2 n+1)^{2}}<\frac{(2n+1)^2}{\pi^2}\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{n(2 n+2) \pi^{2}}{3(2 n+1)^{2}}$$ Now, we can multiply through by \(\pi^2\) to obtain the desired inequality.

4. Use the Squeeze Theorem to prove the final result

We apply the Squeeze Theorem which states that if: $$a_n \le b_n \le c_n$$ and $$\lim_{n\to\infty} a_n = \lim_{n\to\infty} c_n = L,$$ then $$\lim_{n\to\infty} b_n = L$$ From Step 3, we have: $$\frac{n(2 n-1) \pi^{2}}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{n(2 n+2) \pi^{2}}{3(2 n+1)^{2}}$$ Taking limits as n approaches infinity: $$\lim_{n\to\infty} \frac{n(2 n-1) \pi^{2}}{3(2 n+1)^{2}} = \lim_{n\to\infty} \frac{n(2 n+2) \pi^{2}}{3(2 n+1)^{2}} = \frac{\pi^2}{6}$$ Now, applying the Squeeze Theorem, we conclude that: $$\lim_{n\to\infty} \sum_{k=1}^{n} \frac{1}{k^{2}} = \frac{\pi^2}{6}$$ Thus, the infinite sum is: $$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$$

Key concepts

Euler's Formula

Euler's Formula is a fundamental equation in the field of complex analysis and is often regarded as one of the most beautiful mathematical formulas. It demonstrates a deep connection between exponential functions and trigonometric functions. Euler's Formula is defined by the equation \( e^{ix} = \cos(x) + i\sin(x) \), where \( e \) is the base of the natural logarithm, \( i \) is the imaginary unit, and \( \cos(x) \) and \( \sin(x) \) are the trigonometric functions cosine and sine, respectively.
Euler used his formula to demonstrate various mathematical theorems, providing insight into the relationship between numbers and shapes. It plays a crucial role in the study of Fourier transforms and signal processing. Additionally, it is used in the derivation of some remarkable series like the notorious Basel problem, \( \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6} \). Euler's deep insights using complex numbers helped reveal the beauty and coherence of mathematics.
For students diving into the world of complex numbers, Euler’s Formula acts as a bridge connecting algebra, geometry, and analysis. Understanding it can help unravel many complex phenomena, allowing one to appreciate the elegance of mathematics.

Squeeze Theorem

The Squeeze Theorem is a valuable tool in calculus for evaluating limits. It’s especially useful in cases where direct computation of a limit is difficult or when the function does not offer immediate insights into its behavior. The theorem states that if \( a_n \leq b_n \leq c_n \) for all \( n \) and if \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L \), then \( \lim_{n \to \infty} b_n = L \) as well.
This concept was applied in the provided exercise to evaluate the sum \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) by "squeezing" it between two simpler expressions whose limits are more accessible. By strategically finding bounding sequences for the series we're interested in, the Squeeze Theorem enables us to conclude the desired limit is indeed \( \frac{\pi^2}{6} \).
When studying calculus, the Squeeze Theorem is indispensable for proving convergence, particularly for oscillating or bounded sequences that don’t readily lend themselves to straightforward limit computation. It acts like a mathematical vise, helping to pinpoint a limit precisely by surrounding it with two known functions.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is one of the primary trigonometric functions and is defined as the reciprocal of the tangent function, \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \). It is often useful in various domains of mathematics, including calculus and trigonometric identities.
In the context of infinite series, cotangent can be employed for approximations and bounds, as in the inequality given in the exercise, \( \cot^2(x) < \frac{1}{x^2} < 1 + \cot^2(x) \). This inequality serves as a critical component in bounding a series, allowing further mathematical investigation into its convergence.
The significance of the cotangent function extends beyond elementary mathematics into more advanced topics, such as Fourier analysis, where its interplay with periodic functions is explored. Understanding cotangent and its related identities equips students with essential tools for manipulating and analyzing mathematical expressions, ultimately simplifying complex problems.