Question

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

Short answer

\( \sum_{n=1}^{\infty} \frac{1}{n} \).

Step-by-step solution

Recognize the Series Form

Identify the general pattern of the series given: The sequence of numbers given is 1, \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), and so on. This sequence represents the general term of the series as \( \frac{1}{n} \) where \( n \) is the positive integer starting from 1.

Set Up the Sigma Notation

Write the series using sigma notation.The general term \( \frac{1}{n} \) can be expressed in terms of sigma notation as follows: \[\sum_{n=1}^{\infty} \frac{1}{n}\]This sigma notation represents the infinite sum of the sequence \( \frac{1}{n} \) for all positive integers \( n \).

Confirm Your Setup

Ensure that the sigma notation captures the elements of the infinite series correctly.The sigma notation \( \sum_{n=1}^{\infty} \frac{1}{n} \) accurately captures all terms \( (1, \frac{1}{2}, \frac{1}{3}, \ldots) \) as noticed from the given sequence, confirming the notation is correct.

Key concepts

Infinite Series

In an infinite series, a sequence of numbers extends indefinitely. Imagine a never-ending list, like the sequence from 1, 1/2, 1/3, 1/4, and so on. With infinite series, we aim to find the sum of all these numbers, even though there are infinitely many terms. However, the idea is not just about adding numbers perpetually but understanding the behavior of the sum as we add more and more terms. When mathematicians talk about the sum of an infinite series, they often refer to whether the series converges to a specific value or not. If it does not approach a finite value, the series diverges, meaning it keeps growing without bounds or oscillates endlessly.

Although the sequence given, 1 + 1/2 + 1/3 + 1/4 + ..., forms the harmonic series, which is famously known to diverge, we seek to understand its representation through sigma notation for clear understanding and analysis.

General Term Identification

Identifying the general term in a sequence is crucial when using sigma notation. Here, in the sequence 1, 1/2, 1/3, 1/4, ..., each term follows a clear pattern, represented by \( \frac{1}{n} \). This simple expression is the general term for our sequence.

• "General term" means the formula used to find any term in the sequence.
• The variable \( n \) is typically used to denote the position or index of the term within the sequence.
• By using \( \frac{1}{n} \), you can find any term; for instance, if \( n = 2 \), you get \( \frac{1}{2} \).

Identifying the general term helps in succinctly expressing the series as an infinite sum, allowing us to work with the sequence easily and mathematically.

Sequence Representation

Using sigma notation is an efficient way to represent sequences. Instead of writing out many terms, mathematicians use this compact form to describe sequences and their sums. The concept might seem complex at first, but sigma notation simply expresses the sum of a series using a concise formula.

In sigma notation, \[ \sum_{n=1}^{\infty} \frac{1}{n} \], the symbol \( \sum \) (sigma) tells us we are summing terms in a series:

• \( n=1 \) indicates the starting index position in the sequence.
• \( \infty \) signifies that the series continues indefinitely.
• \( \frac{1}{n} \) is the general term, dictating the formula for each term based on its position.

This notation is powerful because it captures the essence of an entire infinite series with just a few symbols, making it essential for mathematical analysis and communication in calculus.