Question

Determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=0}^{\infty} \frac{2 n}{n !}$$

Short answer

The series converges by the Ratio Test.

Step-by-step solution

Identify the General Term

The general term of the series is given by \( a_n = \frac{2n}{n!} \).

Apply the Ratio Test

The Ratio Test involves finding the limit of the absolute value of the ratio of consecutive terms: \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \).

Compute Terms Ratio

Compute \( \frac{a_{n+1}}{a_n} \):\[ \frac{a_{n+1}}{a_n} = \frac{\frac{2(n+1)}{(n+1)!}}{\frac{2n}{n!}} = \frac{2(n+1)}{(n+1)!} \cdot \frac{n!}{2n} = \frac{2(n+1)}{(n+1) \cdot n} = \frac{2}{n} \]

Calculate the Limit

Calculate the limiting value: \[ \lim_{{n \to \infty}} \frac{2}{n} = 0 \]

Conclude Based on Ratio Test

Since the limit is 0, which is less than 1, the Ratio Test indicates that the series converges.

Key concepts

Series Ratio Test

The series ratio test is a fundamental technique in series analysis to determine if a given series converges or diverges. It works by examining the ratio of consecutive terms in a series. Using this method, we focus on the series \( \sum a_n \) and compute \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \). This helps in predicting the convergence behavior:

• If the limit \( L < 1 \), the series converges absolutely.
• If \( L > 1 \) or is infinite, the series diverges.
• If \( L = 1 \), the test is inconclusive.

In our given example, the series is \( \sum_{n=0}^{\infty} \frac{2n}{n!} \). By computing this ratio test correctly, we found that the limit equals 0, indicating the series converges. The value being less than 1 confirms convergence by the ratio test.

General Term of a Series

Understanding the general term of a series, noted as \( a_n \), is crucial for applying convergence tests. The general term provides the representative expression for each term in the series based on its position \( n \). For our example, the series \( \sum_{n=0}^{\infty} \frac{2n}{n!} \) has the general term \( a_n = \frac{2n}{n!} \).
This expression involves several mathematical operators, like factorials, which are common in series concerning exponential functions. A factorial \( n! \) represents the product of all positive integers up to \( n \). Identifying this term helps in subsequent calculations, such as computing the test ratio for convergence. It simplifies the process of mathematical examination by establishing a clear pattern for each series element.

Limit of a Sequence

The concept of a limit is pivotal in understanding series convergence. When we talk about the limit of a sequence, we seek to find what value the sequence \( a_n \) approaches as \( n \) becomes very large. Mathematically, this is expressed as \( \lim_{{n \to \infty}} a_n \).
In the context of the ratio test, we specifically evaluate the limit of the ratio of consecutive terms \( \lim_{{n \to \infty}} \frac{a_{n+1}}{a_n} \). This helps determine whether the series converges or not.
In our problem, \( \lim_{{n \to \infty}} \frac{2}{n} = 0 \), because multiplying by \( \frac{1}{n} \) makes the term tend towards zero as \( n \) grows. This understanding forms the basis for declaring the convergence status under various tests, like the ratio test we used here.