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Magazine Chapter 8: Problem 37
Determine whether the sequence is increasing, decreasing or neither. $$a_{n}=\frac{2^{n}}{(n+1) !}$$
Short Answer
Expert verified
The given sequence \(a_{n}=\frac{2^{n}}{(n+1)!}\) is decreasing.
Step by step solution
01
Determine the nth and (n+1)th terms
Determine the nth term and the (n+1)th term of the sequence. Here, our sequence is \(a_{n}=\frac{2^{n}}{(n+1)!}\). So, our nth term is \(a_{n}\) and our (n+1)th term is \(a_{n+1}=\frac{2^{n+1}}{(n+2)!}\).
02
Find the ratio of the terms
Using the (n+1) th term and the nth term, we find the ratio. This is done by \(\frac{a_{n+1}}{a_{n}} = \frac{\frac{2^{n+1}}{(n+2)!}}{\frac{2^{n}}{(n+1)!}}\). Simplifying this, we get the ratio as \(\frac{2}{n+2}\).
03
Analyze the trend of the ratio
Analyze this ratio. It becomes obvious that as n increases, the ratio \(\frac{2}{n+2}\) decreases.
04
Make the final judgement
Since with each step the ratio \(\frac{2}{n+2}\) becomes smaller, we can conclude that the sequence \(a_{n}=\frac{2^{n}}{(n+1)!}\) is decreasing.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Increasing Sequences
An increasing sequence is a series of numbers where each term is greater than or equal to the one before it. For any sequence \(a_n\), it is called increasing if \(a_{n+1} \geq a_n\) for all \(n\). This means that as you move through the sequence, the values either stay the same or become larger. Simple examples of increasing sequences include:
- The sequence \(2, 3, 4, 5, 6\), where each number is greater than the previous one.
- The sequence \(-3, -1, 0, 2\), where the numbers still increase even though they start below zero.
- The sequence \(3, 3, 4, 5\), because although some numbers are the same, the overall trend is non-decreasing.
Decreasing Sequences
A decreasing sequence occurs when each term is less than or equal to the previous term. It's the opposite of an increasing sequence. For any sequence \(a_n\), it is decreasing if \(a_{n+1} \leq a_n\) for all \(n\). The values decline or stay constant as you move through the sequence. Some typical examples are:
- The sequence \(5, 4, 3, 2, 1\), where each number is strictly less than the preceding one.
- The sequence \(7, 5, 5, 2\), where numbers can stay the same or become smaller.
- The sequence \(10, 10, 9, 8\), which shows a non-increasing trend.
Factorials
Factorials are a fundamental concept in mathematics, denoted by the symbol \(!\). The factorial of a positive integer \(n\) is defined as the product of all positive integers less than or equal to \(n\). It's expressed mathematically as \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).Here are some key points about factorials:
- Factorial Growth: Factorials grow extremely fast. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\), and \(10! = 3,628,800\).
- \(0!\) is defined as 1. This is an important concept frequently used in combinatorics and probability.
- Factorials are often used in permutations and combinations, describing the number of ways to arrange or choose objects.
Ratio Test
The ratio test is a powerful tool used in mathematical analysis to determine whether an infinite series converges or diverges. The idea is to look at the ratio of successive terms in a series.The test is formally stated as follows:
- Given a series \(\sum a_n\), consider the ratio \(\frac{a_{n+1}}{a_n}\).
- If \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1\), the series converges.
- If \(L > 1\) or \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \infty\), the series diverges.
- If \(L = 1\), the test is inconclusive.
