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Magazine Chapter 11: Problem 27
\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2} e^{-n}$$
Short Answer
Expert verified
The series converges by the ratio test.
Step by step solution
01
Examine the Series
The series given is \( \sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2} e^{-n} \). This is an infinite series, and we need to determine its convergence or divergence.
02
Apply the Ratio Test
The ratio test suggests that for a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \) and \( L < 1 \), then the series converges. Let's find the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \) where \( a_n = \left(1+\frac{1}{n}\right)^{2} e^{-n} \).
03
Calculate \( a_{n+1} \) and \( a_n\)
For \( a_{n} = \left(1+\frac{1}{n}\right)^{2} e^{-n} \), we have:\[a_{n+1} = \left(1+\frac{1}{n+1}\right)^{2} e^{-(n+1)}\]
04
Form the Ratio \( \left| \frac{a_{n+1}}{a_n} \right| \)
We calculate:\[\frac{a_{n+1}}{a_n} = \frac{\left(1+\frac{1}{n+1}\right)^{2} e^{-(n+1)}}{\left(1+\frac{1}{n}\right)^{2} e^{-n}} = \left( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right)^2 e^{-1}\]
05
Simplify the Ratio for Limit Calculation
We simplify the ratio:\[ \left( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right)^2 e^{-1} \approx \left( \frac{n+2}{n+1} \times \frac{n}{n+1} \right)^2 e^{-1} \approx \left( 1 - \frac{1}{n+1} \right)^2 e^{-1}\]
06
Take the Limit as \( n \to \infty \)
Calculate the limit:\[\lim_{n \to \infty} \left( \left( 1 - \frac{1}{n+1} \right)^2 e^{-1} \right) = (1)^2 e^{-1} = e^{-1}\]Since \( e^{-1} < 1 \), according to the ratio test, the series converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ratio Test
The ratio test is a useful tool to determine the convergence or divergence of an infinite series. It helps break down complex series into simpler components to analyze their behavior. For a series \( \sum a_n \), the ratio test involves calculating the limit \( L \) of the absolute value of the ratio \( \frac{a_{n+1}}{a_n} \) as \( n \) approaches infinity.
- If \( L < 1 \), the series converges.
- If \( L > 1 \) or \( L \) is infinite, the series diverges.
- If \( L = 1 \), the test is inconclusive and other methods must be used.
Infinite Series
An infinite series is the sum of an infinite sequence of terms. It is written as \( \sum_{n=1}^{\infty} a_n \). This kind of series does not have a finite number of terms. Rather, it continues indefinitely. Understanding whether such a series converges or diverges is a fundamental question in calculus.
Infinite series can be categorized into different types:
Infinite series can be categorized into different types:
- Geometric Series: Where each term is a constant multiple of the previous term.
- Harmonic Series: A series of the form \( \sum \frac{1}{n} \), which generally diverges.
- Power Series: Represents functions as a sum of powers of variables, like Taylor and Maclaurin series.
Limit
The concept of a limit is central to the process of evaluating infinite series for convergence. In calculus, a limit describes the behavior of a function or sequence as it approaches a particular point or extends towards infinity.
The limit helps us understand whether a series approaches a finite value or not, providing insight into series behavior:
The limit helps us understand whether a series approaches a finite value or not, providing insight into series behavior:
- For series \( \sum a_n \), we often seek \( \lim_{n \to \infty} a_n \) to understand term behavior at the infinite extent.
- The ratio test leverages limits to assess series convergence, examining \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
- Limits help analyze term ratios, extract constants, and simplify expressions. This simplifies decision-making about series behavior.
Convergence and Divergence
The ideas of convergence and divergence determine whether an infinite series settles to a finite value (converges) or grows indefinitely (diverges).
Understanding the convergence of a series is crucial because it indicates the possibility of representing complex constructs through sums of potentially infinite terms. The principles are:
Understanding the convergence of a series is crucial because it indicates the possibility of representing complex constructs through sums of potentially infinite terms. The principles are:
- Convergence: If the series approaches a finite value as more terms are added, it converges.
- Divergence: If the series continues to grow and does not approach a finite limit, it diverges.
- Using tests like the ratio test can quickly reveal the series' tendency to converge or diverge.
