Problem 14 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{10^{n}}{n !} $$

Step-by-Step Solution

Answer

The given series converges.

Step 1: Identify the Series

Recognize that we need to determine the convergence of the series ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Step 2: Compare with the Exponential Series

Compare the given series with the exponential series: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Step 3: Rewrite the given series

Rewrite the given series as Using the exponential series expansion formula: ewlineewlineewlineewlineewlineewline numeric infinity ewlineewlineewlineewline.

Step 4: Conclude about the Convergence

Since the given series is a constant multiple of the exponential series \(\frac{10}{n!} = \frac{10}{n} = 0\). Therefore, we conclude that the series converges. ewlineewlineewlineewlineewlineewline.

Key Concepts

infinite series

An infinite series is the sum of an infinite sequence of terms. Unlike finite sums, an infinite series doesn't stop. Instead, it keeps going forever. For example, the series given in the exercise: \(\frac{10^{n}}{n !}\)We call the sum of such terms an infinite series because we're adding an endless number of terms.

• Convergence: An infinite series converges if the sum approaches a specific value as we add more and more terms.
• Divergence: An infinite series diverges if the sum does not approach any specific value, even as we add more terms.

To determine if the series converges, we typically use various convergence tests based on the series' properties.

exponential series

The exponential series is a fundamental type of series in mathematics. It helps in understanding many important functions. The general form of an exponential series is: \begin{eqnarray} e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + onumber \ +\frac{x^n}{n!} onumber \ onumber\tag{1}onumber\termendousetaga{onumberonumber}{SUM}{INFINITE}{SPECIFICIn this exercise, we recognize that the given series is of the form: \(\frac{10^n}{n!}\) which resembles the exponential series with \(x=10\)

The key feature of the exponential series is its tendency to converge quickly, meaning that the sum of its terms approaches a particular value even if we go on adding smaller terms forever.

convergence tests

Convergence tests are a set of methods used to determine whether an infinite series converges or diverges. Here are some commonly used tests:

• Ratio Test: Helps to decide if a series converges or diverges by comparing the ratios of successive terms.
• Comparison Test: Involves comparing the given series to a known convergent or divergent series.
• Integral Test: Links series to integrals to check convergence by comparing to improper integrals.

In the given exercise, the series \( \frac{10^{n}}{n!} \) converges because it matches the form of the exponential series: \( \frac{x^n}{n!} \). The exponential series is known to converge for any real number \(x\)