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Chapter 13: Problem 40

Write out the terms of each series. $$\sum_{i=1}^{5} \frac{x}{i}$$

Short Answer

Expert verified
x, \(\frac{x}{2}\), \(\frac{x}{3}\), \(\frac{x}{4}\), \(\frac{x}{5}\)

Step by step solution

01

Identify the Summation Index

The given series is \(\frac{x}{i}\), where the summation index \(i\) ranges from 1 to 5.
02

Substitute the Values of i

Substitute each integer value of \(i\) from 1 to 5 into the expression \(\frac{x}{i}\).
03

Write Each Term Individually

For \(i=1\), the term is \(\frac{x}{1}=x\). For \(i=2\), the term is \(\frac{x}{2}\). For \(i=3\), the term is \(\frac{x}{3}\). For \(i=4\), the term is \(\frac{x}{4}\). For \(i=5\), the term is \(\frac{x}{5}\).
04

Combine the Terms

Combine all the terms to write out the series as: \(x, \frac{x}{2}, \frac{x}{3}, \frac{x}{4}, \frac{x}{5}\).

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Key Concepts

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

series expansion
In mathematics, a series is the sum of the terms of a sequence.
When you expand a series, you write out all those terms individually.
Let's look at the given example: \(\frac{x}{i}\), where i goes from 1 to 5.
This notation tells us to substitute the values of i (1, 2, 3, 4, 5) one by one into the expression \(\frac{x}{i}\).
Then write out each term.

So when expanded, you have:
  • \(i = 1\) : \(\frac{x}{1} = x\)
  • \(i = 2\) : \(\frac{x}{2}\)
  • \(i = 3\) : \(\frac{x}{3}\)
  • \(i = 4\) : \(\frac{x}{4}\)
  • \(i = 5\) : \(\frac{x}{5}\)

Putting it all together, the expanded series is: \(x, \frac{x}{2}, \frac{x}{3}, \frac{x}{4}, \frac{x}{5}\).
index substitution
Index substitution is a key step in solving summation problems.
The index is a variable, usually represented by letters like i, n, or k, that takes on values over a specified range.
In this example, the index is i, ranging from 1 to 5.

To perform index substitution, you:
  • Take each value within the range (here 1, 2, 3, 4, 5)
  • Substitute these values one by one into the general term (here \(\frac{x}{i}\))
  • Write down each resulting value

This step converts the general form \(\frac{x}{i}\) into specific terms like x, \frac{x}{2}, \frac{x}{3}, etc.
This makes the series clearer and easier to understand.
fractional terms
Fractional terms are commonly found in summation notation exercises.
They can sometimes be tricky but here's a way to think about them.
A fractional term generally has a numerator and a denominator.
In our example, the numerator is constant (x) and the denominator is the changing summation index (i).

So, as i changes values from 1 to 5, the denominator changes and gives us different fractions:
  • When i = 1, the term is \(\frac{x}{1} = x\)
  • When i = 2, the term is \(\frac{x}{2}\)
  • When i = 3, the term is \(\frac{x}{3}\)
  • When i = 4, the term is \(\frac{x}{4}\)
  • When i = 5, the term is \(\frac{x}{5}\)
This shows how the term value changes with the index.
Understanding fractional terms helps in correctly expanding and solving summations.

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