Problem 5 - Calculus Early Transcendentals

Question

Define infinite series and give an example.

Step-by-Step Solution

Answer

Answer: An infinite series is the sum of the elements of an infinite sequence, denoted by \(S = \sum_{n=1}^{\infty} a_n\) where \(a_n\) is the nth term of the infinite sequence. A concrete example is the geometric series \(\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n\), which can be expressed as the series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots\). The series converges because the common ratio is between -1 and 1.

Step 1: Define Infinite Series

An infinite series is the sum of the elements of an infinite sequence. Mathematically, an infinite series can be represented as the limit of partial sums of an infinite sequence as the number of terms approaches infinity. The infinite series is denoted by \(S = \sum_{n=1}^{\infty} a_n\), where \(a_n\) is the nth term of the infinite sequence.

Step 2: Provide an example

A classic example of an infinite series is the geometric series. Consider the infinite geometric series, where the first term is 1 and the common ratio is \(\frac{1}{2}\). It can be represented as \(\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n\).

Step 3: Expand the series

Write out the first few terms to help visualize the example: \[1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots\]

Step 4: Introduce convergence concept (optional)

An important concept related to infinite series is convergence, which is a criterion to determine if the sum of the series exists or not. In the case of our example, the geometric series converges because the common ratio is between -1 and 1.

Key Concepts

Partial Sums

In the realm of mathematics, especially when delving into infinite series, the concept of partial sums is a pivotal one. A partial sum is essentially a snapshot of the total sum up to a certain number of terms. Imagine lining up dominos; each configuration before they all fall is like a partial sum.

More formally, for a given series, \( S \), the partial sum, \( S_n \), is the sum of the first \( n \) terms of the series. Therefore, \( S_n = a_1 + a_2 + ... + a_n \). As \( n \) grows, we get different partial sums, and observing their behavior as \( n \) approaches infinity can be quite telling. It is this progression of partial sums that leads us to understand whether an infinite series converges to a specific value or diverges, shooting off to infinity.

For example, in our geometric series, the partial sum for the first four terms would be \( S_4 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8} \). By examining these snapshots, or partial sums, we inch closer to grasping the full picture of an infinite series.

Geometric Series

A geometric series is a special type of infinite series that can often be more intuitive and easier to work with compared to other series.

What sets a geometric series apart is its structure: each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \). The series takes the form \( S = a + ar + ar^2 + ar^3 + ... \), where \( a \) is the first term. If the absolute value of the common ratio is less than 1, the series converges; otherwise, it diverges.

To illustrate, the series \( \sum_{n=1}^{\infty} a \cdot r^{n-1} \) with \( a = 1 \) and \( r = \frac{1}{2} \) is a geometric series. The beauty of such a series is in its predictability – knowing just the first term and the common ratio can help in determining the entire series. This predictability also enables one to find a simple formula for the sum of a convergent geometric series, which is \( S = \frac{a}{1-r} \) where \( |r| < 1 \) and \( a \) is the first term.

Convergence

The concept of convergence is a central theme when discussing infinite series. A series converges if the sequence of its partial sums approaches a fixed value as the number of terms goes to infinity. If there's no such fixed value, we say the series diverges.

Imagine walking towards a wall. If with every step you halve the distance to the wall, you are converging to the wall; you'll get closer but never quite hit it. This is analogous to a convergent series: the partial sums get nearer to a certain value but may not actually reach that value within a finite number of terms.

Our geometric series example is a hallmark of a convergent series because the absolute value of its common ratio, \( \frac{1}{2} \), is less than 1. This gives us confidence that the sequence of partial sums will stabilize towards a specific number, allowing the notion of the sum of an infinite series to make practical sense.

Sequence

A sequence is a list of numbers in a specific order. Each number in the sequence is called a term. In many ways, sequences are the foundation upon which series are built. While a series is concerned with the sum of its terms, a sequence is purely about the order and value of each individual term.

Sequences can be finite or infinite, and they can follow simple or complex patterns. The notation for a sequence usually involves \( a_n \), indicating the nth term. For example, the simple sequence \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ... \) shows the terms halving each time, and the term \( a_n = \frac{1}{2^{n-1}} \) would describe the nth term of this particular sequence.

Recognizing the nature of the sequence at hand is the first step in understanding the behavior of the series it generates. Whether we're studying arithmetic progressions or elaborating on more complex patterns, the orderly structure of sequences is a cornerstone of mathematical inquiry.