Question

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

Short answer

The series converges.

Step-by-step solution

Recognize the Series Type

The given series \( \sum_{n=1}^{\infty} \frac{10}{n!} \) is a series where the terms are in the form \( \frac{a}{n!} \), specifically with \( a = 10 \). This structure suggests it might relate to the exponential function series.

Recall the Exponential Series

The exponential series for \( e^x \) is given by:\[\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x.\]This series is convergent for all real numbers \( x \). Here \( x = 10 \), but since \( 10 \) is a constant, we can compare the structure of both series.

Determine the Convergence

Since each term of our given series \( \frac{10}{n!} \) resembles the exponential function series and exponential series converges for any real \( x \), the series \( \sum_{n=1}^{\infty} \frac{10}{n!} \) must also converge.

Key concepts

Factorial Series

Factorial series involve terms where the factorial function, denoted as "!", is part of the denominator, like in the expression \( \frac{1}{n!} \). Let's break down what a factorial is: for any non-negative integer \( n \), the factorial, represented \( n! \), is the product of all positive integers less than or equal to \( n \). For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). This rapid growth in the factorial values is a crucial feature. As \( n \) becomes larger, \( n! \) grows very quickly, making each successive term in a factorial series smaller, especially when compared to any constant numerator.

In our series \( \sum_{n=1}^{\infty} \frac{10}{n!} \), the terms shrink rapidly as \( n \) increases, due to the factorial in the denominator outgrowing the constant numerator. This characteristic is pivotal when analyzing convergence of such series.

Exponential Series

The exponential series is a classic example in mathematics, particularly important due to its relationship with the exponential function \( e^x \). This series is expressed as \( \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \). It's known for converging for all real values of \( x \).

Here's a deeper look into how it works:

• Each term in the exponential series has the form \( \frac{x^n}{n!} \), where \( x \) is usually real number.
• The terms start from \( n = 0 \), thus beginning with \( \frac{x^0}{0!} = 1 \).
• Like factorial series, numeral growth in denominators \( n! \) makes terms smaller very quickly as \( n \) increases.

In our given series, \( x \) is not raised to a power but is a constant 10, which aligns it with the exponential series structure for convergence verification.

Infinite Series

Infinite series are summations that continue endlessly. They proceed by adding an infinite number of terms together, like \( \sum_{n=1}^{\infty} a_n \). The key aspect about infinite series is determining whether they converge (approach a finite value) or diverge (grow indefinitely).

Understanding convergence or divergence involves checking the behavior of the terms as \( n \) grows larger. The rules and tests, such as the Comparison Test, Ratio Test, and specifically for this context, recognizing the pattern of exponential series, help verify if an infinite series converges.

In our problem, since the terms follow a structure similar to \( \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \), knowing the exponential series converges for any real \( x \) reassures us that our factorial-based series \( \sum_{n=1}^{\infty} \frac{10}{n!} \) indeed converges as well.