Problem 12
Question
\(9-12=\) Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $$ \sum_{n=1}^{\infty}(-1)^{n-1} n e^{-n} \quad( | \text { error } |<0.01) $$
Step-by-Step Solution
Verified Answer
The series converges. Use 4 terms to estimate the sum with error < 0.01.
1Step 1: Identifying the Series
The given series is \( \sum_{n=1}^{\infty}(-1)^{n-1} n e^{-n} \). It is an alternating series, with the general term \( a_n = n e^{-n} \).
2Step 2: Applying the Alternating Series Test
According to the Alternating Series Test, a series \( \sum (-1)^{n-1} a_n \) converges if the sequence \( a_n \) is decreasing and \( \lim_{n \to \infty} a_n = 0 \). Here, \( a_n = n e^{-n} \).
3Step 3: Checking the Limit of the General Term
Calculate \( \lim_{n \to \infty} n e^{-n} \). Since \( e^{-n} \) decreases exponentially faster than \( n \) increases, \( \lim_{n \to \infty} n e^{-n} = 0 \). This confirms a necessary condition for convergence.
4Step 4: Showing the Decreasing Nature of Terms
Find the derivative of \( f(n) = n e^{-n} \) to show it's decreasing for large \( n \). The derivative \( f'(n) = e^{-n} (1-n) \) is negative for \( n > 1 \), confirming that \( a_n = n e^{-n} \) is a decreasing sequence.
5Step 5: Finding the Required Number of Terms for Accuracy
For an alternating series converging to \( S \), if \( S_N \) is the partial sum of the first \( N \) terms, the error \( |S - S_N| \leq |a_{N+1}| \). We need \( |a_{N+1}| < 0.01 \). Solve \( (N+1)e^{-(N+1)} < 0.01 \) to find \( N \).
6Step 6: Solving the Inequality Numerically
Test values starting from \( N = 1 \) until the condition \( (N+1)e^{-(N+1)} < 0.01 \) is met. This can be done using a simple computational calculator or software.
Key Concepts
Convergent SeriesError Estimate in SeriesAlternating Series
Convergent Series
A series is called convergent if the sequence of its partial sums approaches a specific finite number as more and more terms are added. In simpler terms, as you keep adding more terms, the overall sum gets closer and closer to a certain value.
For a series given by the sum of its terms, if this sum does not settle at a specific number as the number of terms grows infinitely, the series is divergent.
The concept of convergence is crucial because it ensures that the infinite sum has a well-defined total, or limit. This can be particularly important in applications where infinite processes need to produce finite results. In the given exercise, we analyze the convergence using the Alternating Series Test which guarantees that when certain conditions are met, the series converges.
For a series given by the sum of its terms, if this sum does not settle at a specific number as the number of terms grows infinitely, the series is divergent.
The concept of convergence is crucial because it ensures that the infinite sum has a well-defined total, or limit. This can be particularly important in applications where infinite processes need to produce finite results. In the given exercise, we analyze the convergence using the Alternating Series Test which guarantees that when certain conditions are met, the series converges.
Error Estimate in Series
In practical calculations, we often can't compute an infinite series entirely, so it's necessary to estimate how many terms are needed to achieve a desired accuracy.
When an alternating series converges, the error in approximating the sum by the first few terms is less than the absolute value of the first unused term. This means for a convergent alternating series, if the partial sum is close enough to the real sum, we can confidently stop adding more terms.
For the series in the exercise, the error after summing the first \( N \) terms is at most \( |a_{N+1}| \). We needed this error to be less than 0.01, guiding us in choosing how many terms to compute.
When an alternating series converges, the error in approximating the sum by the first few terms is less than the absolute value of the first unused term. This means for a convergent alternating series, if the partial sum is close enough to the real sum, we can confidently stop adding more terms.
For the series in the exercise, the error after summing the first \( N \) terms is at most \( |a_{N+1}| \). We needed this error to be less than 0.01, guiding us in choosing how many terms to compute.
Alternating Series
An alternating series is characterized by terms that switch sign between positive and negative. This can be represented as a series of the form \( \sum (-1)^n a_n \) where the terms \( a_n \) are all positive.
For an alternating series to converge, two conditions must be met. First, the absolute value of the terms \( a_n \) must decrease steadily. Second, \( \lim_{n \to \infty} a_n \) must equal zero.
In the exercise, the given series is an example of an alternating series with \( a_n = n e^{-n} \). It is shown that \( a_n \) decreases and satisfies these conditions through evaluating the limit of \( n e^{-n} \), which approaches zero as \( n \) becomes very large. By testing these conditions, we assure convergence and determine terms needed to approximate the sum with desired accuracy.
For an alternating series to converge, two conditions must be met. First, the absolute value of the terms \( a_n \) must decrease steadily. Second, \( \lim_{n \to \infty} a_n \) must equal zero.
In the exercise, the given series is an example of an alternating series with \( a_n = n e^{-n} \). It is shown that \( a_n \) decreases and satisfies these conditions through evaluating the limit of \( n e^{-n} \), which approaches zero as \( n \) becomes very large. By testing these conditions, we assure convergence and determine terms needed to approximate the sum with desired accuracy.
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