Problem 4 - Calculus Single Variable

Question

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{10^{n}}{n !}\)

Step-by-Step Solution

Answer

The series converges.

Step 1: Understand the Ratio Test

The Ratio Test is a method to determine the convergence of an infinite series. To apply it, calculate the limit \[ L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|. \]If \( L < 1 \), the series converges absolutely. If \( L > 1 \), the series diverges. If \( L = 1 \), the test is inconclusive.

Step 2: Identify the terms of the series

For the series \( \sum_{n=1}^{\infty} \frac{10^{n}}{n!} \), the general term \( a_n \) is \( \frac{10^{n}}{n!} \). We will need \( a_{n+1} \), which is \( \frac{10^{n+1}}{(n+1)!} \).

Step 3: Calculate the ratio \( \frac{a_{n+1}}{a_n} \)

Calculate the ratio of consecutive terms:\[\frac{a_{n+1}}{a_n} = \frac{10^{n+1}}{(n+1)!} \times \frac{n!}{10^n} = \frac{10 \cdot 10^n \cdot n!}{(n+1) \cdot 10^n \cdot n!} = \frac{10}{n+1}.\]

Step 4: Take the limit of the ratio as \( n \) approaches infinity

Now, find the limit:\[L = \lim_{{n \to \infty}} \left| \frac{10}{n+1} \right| = \lim_{{n \to \infty}} \frac{10}{n+1} = 0.\]Since this limit is \( 0 \), which is less than \( 1 \), the series converges.

Key Concepts

Convergence of Series

When we talk about the convergence of a series, we essentially ask whether adding up all the terms in an infinite series results in a finite number or not. This is crucial because if a series converges, it has a specific sum, and our calculations won't go haywire if we try to sum them.
The Ratio Test, which was used in this exercise, is one of the tools to determine whether a series converges or diverges:

• If the limit of the ratio of successive terms is less than 1, the series converges.
• If it is greater than 1, the series diverges.
• However, if the limit equals 1, the test is inconclusive and other methods must be used.

For the provided series, because the ratio limit was 0, the series converges. This means that the infinite sum of this series equals a specific number, making it stable and predictable.

Infinite Series

An infinite series is simply the sum of an infinite sequence of numbers. In mathematical notation, it's usually shown with a summation sign (∑).
For example, in our exercise, the series \[\sum_{n=1}^{\infty} \frac{10^{n}}{n!}\] denotes the sum of terms starting from when \(n = 1\) and continuing indefinitely.
The beauty of infinite series is that although they have an endless number of terms, they can sometimes sum to a finite number if they converge.
In the context of this problem, each term gets progressively smaller because of the division by \(n!\) (factorial of \(n\)), leading to the convergence of the series when assessed with the Ratio Test.

Factorials

Factorials are a mathematical operation where a number is multiplied by all the positive integers less than it.
It's denoted with an exclamation mark, like this: \(n!\). For instance, \(3! = 3 \times 2 \times 1 = 6\).
Factorials grow very large, very quickly.

• \(0!\) is defined as \(1\).
• \(n! = n \times (n-1) \times (n-2) \times \, ... \, \times 2 \times 1\)

In the series we were examining, each term in the series is divided by \(n!\).
This means that as \(n\) increases, the denominator becomes significantly larger, causing both the terms in the series to become smaller rapidly and contributing to the convergence of the infinite series.

Limit of Sequence

In mathematics, taking the limit of a sequence means finding what value a function or sequence approaches as the index (in this case \(n\)) goes to infinity.
This is a key idea when applying the Ratio Test.For the series \[a_n = \frac{10^{n}}{n!}\] and \[a_{n+1} = \frac{10^{n+1}}{(n+1)!},\]We computed the ratio of consecutive terms, \[\frac{a_{n+1}}{a_n} = \frac{10}{n+1}.\] Then, by taking the limit as \(n\) approaches infinity, \[L = \lim_{{n \to \infty}} \frac{10}{n+1} = 0.\] This result, being less than 1, confirms that the series converges.
Understanding limits provides us with a final answer on a sequence's behavior, ensuring we know exactly how it will act as it extends towards infinity.