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Chapter 12: Problem 55

Expand the partial sum and find its value. \(\sum_{k=0}^{3} \frac{4}{k !}\)

Short Answer

Expert verified
10.6667

Step by step solution

01

Identify the series and range

The given series is \(\frac{4}{k!}\) and the summation range is from \(k = 0\) to \(k = 3\). This means sum the values of \(\frac{4}{k!}\) for \(k = 0, 1, 2, 3\).
02

Evaluate the factorial values

Calculate the factorial for each value of \(k\): \(0! = 1, 1! = 1, 2! = 2, 3! = 6\).
03

Substitute and simplify each term

Substitute the values of \(k\) into the series and calculate each term: \(\frac{4}{0!} = 4\), \(\frac{4}{1!} = 4\), \(\frac{4}{2!} = 2\), and \(\frac{4}{3!} \approx 0.6667 \).
04

Sum all the terms

Add up the simplified terms: \(4 + 4 + 2 + 0.6667 = 10.6667\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Summation
Summation is a mathematical operation that adds together a sequence of numbers. In this exercise, we are summing the values of a series defined by the term \(\frac{4}{k!}\) from \(k = 0\) to \(k = 3\).
Steps involved in summation:
  • Identify the range: Here, \(k\) ranges from 0 to 3.
  • Evaluate the individual terms in the series within this range.
  • Add these terms together to get the final result.
Summation is often represented using the sigma notation \(\sum\), which makes it compact and readable. For example, \( \sum_{k=0}^{3} \frac{4}{k!} \) is the summation of the series \( \frac{4}{k!} \) from \( k=0 \) to \( k=3 \).
Factorials
Factorials, denoted by \( ! \), are a product of all positive integers up to a certain number. They are fundamental in permutations, combinations, and many areas of mathematical analysis. For instance, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
In our exercise, we need to calculate factorials for \(k = 0, 1, 2, 3 \):
  • \( 0! = 1 \)
  • \( 1! = 1 \)
  • \( 2! = 2 \)
  • \( 3! = 6 \)
These values are then used to simplify each term in the series. Knowing how to compute factorials is essential for evaluating many types of mathematical expressions and series.
Series Expansion
Series expansion refers to the method of expressing a function as a sum of terms. This is common in calculus, especially in understanding functions near a point, and using infinite sums like Taylor or Maclaurin series.
In our example, the series is expressed as \(\sum_{k=0}^{3} \frac{4}{k!}\). To expand this, we follow these steps:
  • Calculate each term: \( \frac{4}{0!} = 4, \frac{4}{1!} = 4, \frac{4}{2!} = 2, \frac{4}{3!} \approx 0.6667 \)
  • Simplify each term: You replace factorial values to get these results.
  • Sum all terms: Adding them gives the partial sum \(4 + 4 + 2 + 0.6667 \approx 10.6667 \).
Series expansions help approximate complex functions and understand them better in simpler forms. It鈥檚 a crucial tool in various fields like physics, engineering, and economics.

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