Question

36\. Replace \(x\) by \(i x\) in the expansion in the text for \(\left(e^{x}-\right.\) \(\left.e^{-x}\right) / 2\) to get both the power series for the sine and a representation of the sine as an infinite product. By using the relationship between the roots and coefficients of a polynomial (extended to power series), show that $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \text { and } \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90} $$

Short answer

Question: Prove the relationship between the sums of reciprocals of squares and fourth powers using power series and infinite product representations of the sine function: $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ and $\sum_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$.

Step-by-step solution

Replace x with ix

Replace x with ix in the expression \(\frac{e^x - e^{-x}}{2}\): $$ \frac{e^{ix} - e^{-ix}}{2} $$

Express the complex exponentials as sinusoidal functions

Use Euler's formula to express the complex exponentials as a combination of sine and cosine functions: $$ \frac{\cos(x) + i\sin(x) - \cos(-x) - i\sin(-x)}{2}=\frac{2i \sin x}{2} $$ Simplify the expression to get the sine function: $$ i \sin(x) $$

Power series representation of sine function

Recall the power series expansion for sine function: $$ \sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!}x^{2k+1} $$

Infinite product representation of sine function

Recall the infinite product representation for the sine function using the Weierstrass product formula: $$ \sin(x) = x\prod_{k=1}^{\infty}\left(1-\frac{x^2}{\pi^2k^2}\right) $$

Use roots-coefficients relationship

From the power series representation, we notice that the coefficient of \(x^2\) is 0 and the coefficient of \(x^4\) is \(\frac{-1}{3!}\). Using the infinite product representation, we can relate these coefficients to the sum of reciprocal of squares and fourth powers. For the coefficient of \(x^2\) from the product representation, we notice that the sum of all terms having \(x^2\) is: $$ \sum_{k=1}^{\infty}\left(-\frac{x^2}{\pi^2k^2}\right) = 0 $$ Multiplying by \(-\pi^2\), we get: $$ \sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6} $$ For the coefficient of \(x^4\) from the product representation, the sum of all terms having \(x^4\) is: $$ \sum_{k=1}^{\infty} \sum_{j=1}^{k-1} \left(-\frac{x^4}{\pi^4k^2j^2}\right) = \frac{-1}{3!} $$ Multiplying by \(-\frac{\pi^4}{4!}\), we have: $$ \sum_{k=1}^{\infty} \sum_{j=1}^{k-1} \frac{1}{k^2j^2} = \frac{\pi^4}{90} $$ which can be rearranged as $$ \sum_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90} $$

Key concepts

Euler's formula

Euler's formula is a profound relationship in mathematics that connects complex exponentials to trigonometric functions. This formula states that for any real number x, the exponential function eix, where i is the imaginary unit, is equivalent to cos(x) + i*sin(x). In essence, Euler's formula translates the complex exponential eix into a complex number on the unit circle in the complex plane.

This discovery is vital for understanding the behavior of waves, oscillations, and even quantum mechanics. It enables the conversion between exponential and trigonometric representations, which is incredibly useful in calculus and differential equations. Applying this formula simplifies many problems in mathematics, physics, and engineering.

Infinite product representation

The infinite product representation is another fascinating aspect of trigonometric functions like the sine function. This representation tells us that functions can be constructed as products involving their zeros. For the sine function, it is represented as sin(x) = x*∏k=1∞(1 - x2/π2k2).

The sine function has zeros at every integer multiple of π, and this formula neatly encapsulates that property. By recognizing these zeros, the formula provides a clear picture of the function's behavior over its entire domain. It's notably used in areas such as complex analysis and number theory.

Roots-coefficients relationship

The relationship between the roots and coefficients of a polynomial is an essential concept in algebra, which extends to power series in the context of calculus. The sums of the reciprocals of the squares and fourth powers of integers can be linked to the coefficients of the power series of trigonometric functions like sine.

The coefficients give us insights into the behavior of the function at different powers of x, and by setting certain terms equal to zero or relating them to other known quantities, we can uncover significant mathematical constants such as π²/6 or π⁴/90, as seen in the given exercise. These relationships reveal profound connections within mathematics, tying together seemingly disparate areas through the properties of functions and series.