Problem 38 - Calculus

Question

By recognizing each series as a Taylor series evaluated at a particular value of \(x,\) find the sum of each of the following convergent series. $$1-\frac{100}{2 !}+\frac{10000}{4 !}+\cdots+\frac{(-1)^{n} \cdot 10^{2 n}}{(2 n) !}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \cos(10) \).

Step 1: Identify the General Term

The given series is: \( 1 - \frac{100}{2!} + \frac{10000}{4!} + \cdots + \frac{(-1)^{n} \cdot 10^{2n}}{(2n)!} + \cdots \) is of the form \( \sum_{n=0}^{\infty} \frac{(-1)^{n} \cdot a^{2n}}{(2n)!} \).

Step 2: Recognize the Series as a Taylor Series

Recognize that the series \( \sum_{n=0}^{\infty} \frac{(-1)^{n} \cdot a^{2n}}{(2n)!} \) matches the Taylor series expansion for the function \( \cos(x) \) where the function \( \cos(x) \) is expanded at x = 0, and \( x^n \) is substituted by \( a^{2n} \).

Step 3: Identify the Correct Function Parameters

For \( \sin(x) \), the series expansion is \( \sum_{n=0}^{\infty} \frac{(-1)^{n} \cdot x^{2n}}{(2n)!} \). Here, \( x^2 = 100 \), so \( x = 10 \).

Step 4: Evaluate the Series

Substitute \( x = 10 \) in the function \( \cos(x) \). This gives \( \cos(10) \).

Step 5: Determine the Sum

Since the series converges to \( \cos(10) \), the sum of the series is: \( \cos(10) \). You may use a calculator to find \( \cos(10) \) for an approximate decimal value if necessary.

Key Concepts

Cosine function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function. It describes the control of rotation on the unit circle. For each angle \( x \), \( \cos(x) \) represents the x-coordinate of the point on the unit circle. The function is periodic with a period of \( 2\pi \), meaning every \( 2\pi \) units, the cosine wave repeats its cycle.
Cosine is vital because of its relationships with other trigonometric functions and its real-world applications in waves and oscillations. Key properties include:

• The range of \( \cos(x) \) is \([-1, 1]\).
• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• It achieves its maximum value at even multiples of \( \pi \), and the minimum value at odd multiples of \( \pi \).

Understanding \( \cos(x) \) provides insight into many mathematical and physical phenomena, making it a crucial aspect of trigonometry.

Series convergence

Series convergence refers to the behavior of a series as the number of terms approaches infinity. A series converges if the sum of its terms approaches a finite limit. Otherwise, it diverges.
For a series \( \sum_{n=0}^{\infty} a_n \) to converge:

• The sequence of partial sums \( S_N = a_1 + a_2 + \cdots + a_N \) must approach a specific value as \( N \to \infty \).
• This means \( S_N \) gets closer and closer to some finite limit, known as the sum of the series.
• Techniques like the Ratio Test, Root Test, and Comparing Series Tests are often used to establish convergence.

In the context of Taylor series, analyzing the convergence is essential to ensure the expanded function accurately reflects the behavior of the function it represents over an interval.

Taylor series expansion

A Taylor series is a powerful mathematical tool that expresses a function as an infinite sum of terms calculated from the function's derivatives at a specific point. This technique is often utilized for approximations.
The general form of a Taylor series for a function \( f(x) \) at a point \( a \) is:\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]where \( f^{(n)}(a) \) denotes the \( n \)-th derivative of \( f(x) \) evaluated at \( a \).
For the cosine function centered at zero, the Taylor series becomes:\[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \]This series converges for all real numbers due to its properties as an entire function. Taylor series allow us to evaluate complex functions easier by turning them into polynomial-like expressions that are often more manageable.