Problem 36 - 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. $$1-\frac{1}{3 !}+\frac{1}{5 !}-\frac{1}{7 !}+\dots+\frac{(-1)^{n}}{(2 n+1) !}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \(\sin(1)\).

Step 1: Identify the pattern

First, recognize the pattern in the series: the terms alternate in sign and involve odd factorials in the denominator. The given series is: \[ 1 - \frac{1}{3!} + \frac{1}{5!} - \frac{1}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}. \]

Step 2: Recall the Taylor series for sine

Recall the Taylor series expansion for \(\sin(x)\) centered at 0 is: \[ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}. \]

Step 3: Compare the series and Taylor series

Notice that the given series resembles the Taylor series for \(\sin(x)\), but without the \(x^{2n+1}\) term in each component. That is, for \(\sin(x)\) with \(x = 1\), it becomes:\[ \sin(1) = 1 - \frac{1}{3!} + \frac{1}{5!} - \frac{1}{7!} + \cdots. \]

Step 4: Evaluate the Taylor series at a specific point

Since the series is equivalent to the sum of \(\sin(x)\) with \(x = 1\), the sum of the series is simply \(\sin(1)\). Therefore, \[ \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} = \sin(1). \]

Step 5: Conclusion

The given series evaluates to the sine of 1 radian. Hence, the sum of the series is \(\sin(1)\).

Key Concepts

Convergent Series

A convergent series is a sequence of numbers that approaches a specific value as more terms are added. This means the sum of the series gets closer and closer to a certain number, even though there may be infinitely many terms. A key feature is that despite an infinite number of terms, the overall sum remains finite.

• One example of a convergent series is the geometric series, where the terms form a geometric progression.
• Another example is the alternating harmonic series, where the signs of the terms alternate between positive and negative.

The series presented in the original exercise is a specific example of a convergent series. It is derived from the Taylor series for the sine function evaluated at a specific point. Its convergence means that as you add more and more terms, the sum approaches the value of \( \sin(1) \). It’s important to identify this pattern to understand the nature of the given series.

Sine Function

The sine function is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle as a function of the angle from the positive x-axis. It is an essential component of many mathematical equations and is crucial for modeling periodic phenomena such as sound waves and tidal patterns.

• The sine function is periodic with a period of \(2\cdot\pi\).
• Its range is from -1 to 1, corresponding to the maximum and minimum y-values on the unit circle.
• The sine of an angle is equal to the opposite side divided by the hypotenuse in a right-angled triangle.

In the context of this exercise, the series presented is the Taylor series of the sine function evaluated at \( x = 1 \). This reveals the connection between the sine function and the cumulative sum of a series of fractions with factorial denominators.

Taylor Expansion

Taylor expansion is a method of approximating complex functions with a series of polynomial terms. This technique is extremely useful because it allows us to express a function in terms of its derivatives at a specific point, providing an approximation that becomes increasingly accurate as more terms are included.

• It gives a function’s value at a point \( x \) through the derivatives evaluated at a nearby point, commonly zero.
• Each term of the Taylor series involves derivatives of the function, and these derivatives are divided by factorial terms.
• The Taylor series for \( \sin(x) \) is unique because it only includes odd powers of \( x \).

In this exercise, the Taylor expansion for \( \sin(x) \) is evaluated at \( x = 1 \), forming an infinite series whose sum converges to the sine of 1 radian. This highlights the power of Taylor series in linking seemingly complex infinite series to familiar functions.

Factorials

Factorials are a mathematical function that involves multiplying a series of descending natural numbers. They are critical in combinatorics, calculus, and many areas of mathematical computation.

• A factorial of a non-negative integer \( n \) is denoted as \( n! \) and is defined as the product of all positive integers less than or equal to \( n \).
• By definition, \( 0! = 1 \), which is an important base case.
• Factorials grow very rapidly; for instance, \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \).

In the series from the exercise, each denominator is a factorial of an odd number. Factorials are used in the Taylor series to normalize the growth of powers of \( x \), ensuring the series converges. They act as a balancing factor, dampening the effect of higher powers in the polynomial expansion.