Problem 43 - 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. $$1-0.1+\frac{0.01}{2 !}-\frac{0.001}{3 !}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \(e^{-0.1}\).

Step 1: Identify the General Term

The series given is:\[ 1 - 0.1 + \frac{0.01}{2!} - \frac{0.001}{3!} + \cdots \]The general term of this series can be written as:\[ (-1)^n \frac{(0.1)^n}{n!} \]This suggests that the series is an alternating series involving powers of 0.1.

Step 2: Recognize the Taylor Series

The general term \((-1)^n \frac{(0.1)^n}{n!}\) is reminiscent of the Taylor series for \(e^x\), which is:\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]However, the series here has an alternating sign. This suggests it matches the series for \(e^{-x}\).

Step 3: Determine the Function and Value of x

The series matches the form for \(e^{-x}\) with \(x = 0.1\). Thus, this series is the Taylor series expansion for \(e^{-0.1}\).

Step 4: Evaluate the Sum of the Series

Since the series is the Taylor series representation for \(e^{-0.1}\), its sum is simply the evaluation of the exponential function at \(x = -0.1\). Therefore, the sum is:\[ e^{-0.1} \]

Key Concepts

Convergent Series

A convergent series is a sequence of numbers that approaches a specific value as more and more terms are added. This concept is crucial in understanding complex mathematical series and their applications. When we say a series is convergent, it means that as you add more terms, the sum of the series gets closer to a certain number.

• Think of it as having an infinite number of terms, yet the total remains finite.
• This is opposed to a divergent series, where the sum becomes infinitely large or does not settle on any value.

The Taylor series used in finding such sums is typically convergent within a particular range, known as the radius of convergence. In the case of the series expanding to find the value of functions like the exponential function, convergence allows us to approximate these functions to any desired degree of accuracy by adding more terms.

Exponential Function

The exponential function, commonly denoted as \(e^x\), is a fundamental mathematical function with unique properties that make it crucial in calculus and real-world applications. The function \(e^x\) is defined through its Taylor series expansion:\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]This series is infinite but converges to \(e^x\) for all real numbers \(x\).

• The number \(e\) is approximately 2.71828 and is an irrational number.
• It serves as the base of natural logarithms and has properties like \(e^0 = 1\) and \(e^{x+y} = e^x \, e^y\).

The exponential function grows exceptionally fast and is widely used to model growth processes, like population growth or radioactive decay. In mathematical terms, it is part of the family of exponential functions, which can be shifted or reflected depending on the parameters used.

Alternating Series

An alternating series is a series in which the terms change sign between positive and negative, such as the series example \(1 - 0.1 + \frac{0.01}{2!} - \frac{0.001}{3!} + \cdots\). These series have particularly useful convergence properties.

• In an alternating series, the absolute value of each term is less than the one before it and approaches zero.
• The convergence of an alternating series can be determined by the Alternating Series Test, which states that if the absolute values of the terms are decreasing and approaching zero, the series converges.

Alternating series often appear in Taylor series expansions for functions involving both "+" and "-" terms, like \(e^{-x}\) in this case, leading to convergence within a certain range. This oscillation between terms is what sets alternating series apart from regular series and allows them to accurately approximate functions that alternate in nature.