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Chapter 5: Problem 67

Using sigma notation, write the following expressions as infinite series. $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $$

Short Answer

Expert verified
The series is written as \( \sum_{n=1}^{\infty} \frac{1}{n} \).

Step by step solution

01

Identify the Pattern

Observe the given sequence: \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\). Notice that each term is of the form \(\frac{1}{n}\), where \(n\) is an integer starting from 1.
02

Determine the General Term

From the pattern \( \frac{1}{n} \), we deduce that the general term of the series is \( a_n = \frac{1}{n} \). This general term represents each term in the sequence.
03

Use Sigma Notation

Express the sequence as an infinite series using sigma notation. The series can be written as: \[ \sum_{n=1}^{\infty} \frac{1}{n} \]This represents the infinite sum of the terms in our sequence, starting at \( n = 1 \) and continuing indefinitely.

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Key Concepts

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

sigma notation
Sigma notation is a way of writing a series using the Greek letter "危". It's a concise mathematical way to express the sum of several terms when they follow a specific pattern. In sigma notation, the notation looks like this: \( \sum_{n=a}^{b} f(n) \). This translates to, "Sum of \( f(n) \) terms, starting from \( n = a \) to \( n = b \)."
  • The symbol \( \Sigma \) stands for summation - think of it as a mathematical way to write "add everything up".
  • \( n \) is the index of summation. It tells you which number to start from.
  • \( f(n) \) represents the general formula or rule for generating the terms you want to sum.
In the problem we are solving, because it is an infinite series, there is no upper bound. Thus, we use \( \infty \) as the upper limit, resulting in the expression \( \sum_{n=1}^{\infty} \frac{1}{n} \). This means start from \( n=1 \) and go on forever!
Sigma notation elegantly captures the idea of summing an infinite number of sequence terms.
sequence
A sequence in mathematics is an ordered list of numbers. Each number in the list is called a term. Sequences are defined based on a specific rule that determines the order and value of its terms. Think of a sequence as an instruction that tells you how to go from one number to the next in a list.
  • The sequence given in the exercise is \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \)
  • It's an infinite sequence, meaning it keeps going without end.
  • A sequence can be described using an explicit formula, which tells you how to find any term in the sequence.
Understanding sequences is crucial because series, such as the one in this exercise, are derived from sequences. Each term in a series is essentially a term of a specific sequence.
general term
The general term of a sequence is a formula that allows you to find any term in the sequence. This is a key concept in analyzing sequences and series because it helps pinpoint exactly what each term looks like without needing to list them.
  • The general term is often written as \( a_n \), where \( n \) represents the position of the term in the sequence.
  • For the sequence \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \), the general term is \( a_n = \frac{1}{n} \).
  • The formula tells us that for the nth term, you need to divide 1 by \( n \).
Knowing the general term is essential, especially when using sigma notation, because it provides the specific pattern or formula to sum over.
harmonic series
The harmonic series is a specific type of series where each term is the reciprocal of a positive integer. It's a classic example of an infinite series studied in mathematics.
  • It is represented as \( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \)
  • Each term of the harmonic series takes the form \( \frac{1}{n} \), which is consistent with the general term \( a_n = \frac{1}{n} \).
  • It's called "harmonic" because of its connection to harmonically related frequencies in music.
  • This series diverges. This means that as you add more and more terms, the series as a whole increases without bound rather than settling at a particular value.
The harmonic series is an interesting example used to explore convergence and divergence properties in calculus and beyond.

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Most popular questions from this chapter

If \(\quad b_{n} \geq 0 \quad\) is \(\quad\) decreasing \(\quad\) and \(\sum_{n=1}^{\infty}\left(b_{3 n-2}+b_{3 n-1}-b_{3 n}\right) \quad\) converges then \(\sum_{n=1}^{\infty} b_{3 n-2}\) converges.

Let \(a_{n}=\frac{1}{4} \frac{3}{6} \frac{5}{8} \cdots \frac{2 n-1}{2 n+2}=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2^{n}(n+1) !}\) Explain why the ratio test cannot determine convergence of \(\sum_{n=1}^{\infty} a_{n} .\) Use the fact that \(1-1 /(4 k)\) is increasing \(k\) to estimate \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}\).

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}$$

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum_{k=1}^{\infty} a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$a_{k}=\frac{2 \cdot 4 \cdot 6 \cdots 2 k}{(2 k) !}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=\left(\frac{k}{k+\ln k}\right)^{2 k}$$
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