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Chapter 9: Problem 29

Use any method to determine whether the series converges. \(\sum_{k=1}^{\infty} k^{50} e^{-k}\)

Short Answer

Expert verified
The series converges by the Ratio Test.

Step by step solution

01

Choose a Convergence Test

To determine the convergence of the series \( \sum_{k=1}^{\infty} k^{50} e^{-k} \), we'll use the Ratio Test. This is a suitable test since the series involves exponential terms.
02

Apply the Ratio Test

The Ratio Test involves considering \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). Here, \( a_k = k^{50} e^{-k} \), so \( a_{k+1} = (k+1)^{50} e^{-(k+1)} \). Therefore, \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{50} e^{-(k+1)}}{k^{50} e^{-k}} = \frac{(k+1)^{50}}{k^{50}} e^{-1} \).
03

Simplify and Evaluate the Limit

Simplify \( \frac{(k+1)^{50}}{k^{50}} e^{-1} = \left(1 + \frac{1}{k} \right)^{50} e^{-1} \). The limit becomes \( \lim_{k \to \infty} \left(1 + \frac{1}{k} \right)^{50} e^{-1} = 1 \cdot e^{-1} = \frac{1}{e} \).
04

Conclude with the Ratio Test

Since \( \frac{1}{e} < 1 \), the Ratio Test tells us that the series \( \sum_{k=1}^{\infty} k^{50} e^{-k} \) 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 powerful method used for determining the convergence of infinite series, especially when dealing with terms that involve exponential expressions. To apply the Ratio Test, we examine:
  • For a series with terms \( a_k \), compute the ratio \( \left| \frac{a_{k+1}}{a_k} \right| \).
  • Evaluate the limit as \( k \) approaches infinity, i.e., \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
The outcome of this test provides insight about the series:
  • If the limit is less than 1, the series converges absolutely.
  • If the limit is greater than 1, the series diverges.
  • If the limit equals 1, the test is inconclusive.
For our series \( \sum_{k=1}^{\infty} k^{50} e^{-k} \), employing the Ratio Test resulted in a limit of \( \frac{1}{e} \), which is less than 1, confirming convergence.
Exponential Series
An Exponential Series is one in which the term contains an exponential function, usually of the form \( a^k \) or \( e^{-k} \). These series are prevalent in mathematical problems and can often be tricky due to their rapidly changing nature. Understanding the behavior of exponential terms is crucial:
  • Exponential decay (e.g., \( e^{-k} \)) tends to drive series toward convergence when combined with other decreasing functions.
  • The function \( e^{-k} \) decreases quickly as \( k \) increases, helping terms to shrink toward zero.
For the series \( \sum_{k=1}^{\infty} k^{50} e^{-k} \), the factor \( e^{-k} \) dominates for large \( k \), and despite \( k^{50} \) growing, the exponential decay ensures overall convergence.
Limit Evaluation
Limit Evaluation is a mathematical technique to analyze the behavior of sequences or series as terms approach infinity. It is essential in determining convergence characteristics:
  • We compute the limit of the sequence or series term ratio \( a_k \) as \( k \) becomes very large.
  • This often involves simplifying the expression to a recognizable form where its limit can be evaluated.
In the context of the series given by \( \sum_{k=1}^{\infty} k^{50} e^{-k} \), the essential step was evaluating:
  • \( \lim_{k \to \infty} \left( 1 + \frac{1}{k} \right)^{50} e^{-1} = \frac{1}{e} \).
This step helped demonstrate that the terms shrink to zero sufficiently fast, confirming the series' convergence using the Ratio Test.

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Most popular questions from this chapter

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