Problem 79 - Calculus: Early Transcendentals

Question

Find the sum of the series. \( 3 + \frac {9}{2!} + \frac {27}{3!} + \frac {81}{4!} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The sum of the series is \( e^3 - 1 \).

Step 1: Identify the Pattern

Observe the given series: \( 3 + \frac{9}{2!} + \frac{27}{3!} + \frac{81}{4!} + \, \cdot \cdot \cdot \). Each term is composed of a power of 3 in the numerator and a factorial in the denominator. The general term can be recognized as \( \frac{3^n}{n!} \).

Step 2: Recognize a Known Series

The series resembles the exponential series for \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \). By setting \( x = 3 \), we see that the sum of the series \( \sum_{n=1}^{\infty} \frac{3^n}{n!} \) is related to \( e^3 \).

Step 3: Calculate the Sum of the Series

The given series starts at \( n=1 \): \( \sum_{n=1}^{\infty} \frac{3^n}{n!} \). This can be rearranged using the exponential series for \( e^3 \): \( e^3 = \sum_{n=0}^{\infty} \frac{3^n}{n!} \). Thus, the series \( \sum_{n=1}^{\infty} \frac{3^n}{n!} = e^3 - \frac{3^0}{0!} = e^3 - 1 \).

Step 4: Simplify the Expression

The sum of the series is \( e^3 - 1 \). This is because the series without the first term (for \( n=0 \)) corresponds to the exponential series starting from \( n=1 \).

Key Concepts

Power Series

Power series are fascinating tools for representing functions as infinite sums of terms. In mathematical terms, a power series is an infinite series of the form:\[\sum_{n=0}^{fty} a_n (x - c)^n \]where:

• \( a_n \) represents the coefficients of the series,
• \( x \) is the variable,
• \( c \) is the center of the series, often zero.

Power series allow us to express more complex functions in terms of simpler polynomial-like forms, which makes them handy for calculations and approximations.
In this exercise, the series can be analyzed as a power series where the base is the constant 3, represented through increasing powers \( 3^n \).
This is particularly useful because it resembles the expansion of functions, like the exponential function.
Power series converge differently based on the radius of convergence, an important aspect for determining how and where the series will accurately approximate the function it expresses.

Factorials

Factorials are mathematical expressions that represent the product of a sequence of descending natural numbers. It is denoted by an exclamation mark such as \( n! \).
For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials grow very quickly in size with increasing \( n \), making them crucial for many combinatorial and series calculations.
In the given series, each term's denominator is a factorial, such as \( 2!, 3!, 4! \), etc., which helps in controlling the growth of the terms and ensures their convergence.
Factorials are a key component in defining an exponential series, as they serve to "normalize" the large numbers that result from taking high powers of a base in the numerators.
They are essential in fields like probability, combinatorics, and in various computational problems where permutations and combinations are considered.

Exponential Function

The exponential function is a fundamental mathematical function denoted by \( e^x \), where \( e \) is Euler's number, approximately 2.71828. This function models growth processes, such as population growth, and captures the idea of continuous compounding in finance.
The exponential function has a special property where its derivative is the function itself, a trait that makes it unique and important in calculus and differential equations.
The power series representation of the exponential function is given by:\[e^x = \sum_{n=0}^{fty} \frac{x^n}{n!}\]This series converges for all real (and even complex) values of \( x \), providing a robust framework for analyzing exponential growth and decay processes.
In this exercise, setting \( x=3 \) quickly identifies the series as that of \( e^3 \). Furthermore, by adjusting the starting point of the series to \( n=1 \), we find the sum as \( e^3 - 1 \).
The manipulation and understanding of such series is useful in physics, engineering, and economics, where exponential functions are heavily applied.