Problem 11 - Thomas Calculus

Question

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)$$

Step-by-Step Solution

Answer

The series sum is 11.5.

Step 1: Understanding the Series

The series given is \( \sum_{n=0}^{ot\infty}\left(\frac{5}{2^n}+\frac{1}{3^n}\right) \). It's a combination of two geometric series, \( \sum_{n=0}^{\infty} \frac{5}{2^n} \) and \( \sum_{n=0}^{\infty} \frac{1}{3^n} \). A geometric series is of the form \( \sum_{n=0}^{\infty} ar^n \), and it converges if \( |r| < 1 \).

Step 2: Writing the First Eight Terms

Calculate the first eight terms by substituting \(n = 0, 1, 2, \ldots, 7\) into the formulas for each geometric component:For \(\frac{5}{2^n}\), the terms are \(5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \frac{5}{16}, \frac{5}{32}, \frac{5}{64}, \frac{5}{128}\).For \(\frac{1}{3^n}\), the terms are \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \frac{1}{729}, \frac{1}{2187}\).Combine these terms to get the first eight terms of the initial series.

Step 3: Calculating Sum of the Series

Since both series are geometric with ratios \(r_1 = \frac{1}{2}\) and \(r_2 = \frac{1}{3}\), calculate their sums. For a geometric series \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\):- For \(\sum_{n=0}^{\infty} \frac{5}{2^n}\): \(a = 5, r = \frac{1}{2}\). So, the sum is \(\frac{5}{1-\frac{1}{2}} = 10\).- For \(\sum_{n=0}^{\infty} \frac{1}{3^n}\): \(a = 1, r = \frac{1}{3}\). So, the sum is \(\frac{1}{1-\frac{1}{3}} = \frac{3}{2}\).

Step 4: Finding the Total Sum

Add the sums of the two geometric series obtained in the previous step: \(10 + \frac{3}{2} = \frac{20}{2} + \frac{3}{2} = \frac{23}{2}\) or \(11.5\).This is the sum of the series \(\sum_{n=0}^{\infty}\left(\frac{5}{2^n}+\frac{1}{3^n}\right)\).

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. In mathematics, when we talk about series, specifically infinite series, we're referring to an ongoing sequence that has no end. Imagine listing numbers one after another, forever, and then adding them up.
This might sound impossible at first, but with mathematical tools, we can analyze these never-ending sums.

• Infinite series are often written using the summation symbol \( \sum \).
• Each term in a series is a part of a sequence, usually related by a certain rule or pattern.
• The goal is often to see if this infinite sum adds up to a finite number or if it just keeps getting larger and larger.

Understanding these series is important when examining functions and patterns in advanced mathematics.
In the context of our exercise, assume we're dealing with a series made up from different components, each with its own rule of formation. Analyzing these helps us see the overall story the series is trying to tell.

Convergence of Series

The convergence of a series is vital because it helps us determine whether the infinite series we're dealing with actually sums up to a finite value. Simply put, convergence tells us if we can attach a single number to the "sum" of all the infinite terms.
To see if a series converges, consider these points:

• A series is convergent if it approaches a specific value, just as you were gradually adding more and more tiny bits to reach that value.
• If the series doesn't settle down to a specific number, it's divergent and effectively doesn't "add up" in a classical sense.
• For geometric series like in our exercise, they converge if the common ratio \( r \) fits within -1 and 1 (i.e., \( |r| < 1 \)).

Geometric progressions are particularly friendly when determining convergence, as they provide straightforward conditions using the common ratio. Once we determine convergence, we can further derive the actual sum, leading us to our next concept.

Sum of Series

Finding the sum of a series involves putting together all those infinite terms into just one tidy number. This might sound hard, especially when we're talking about infinity, but certain types of series make this simple.
For our exercise, we've tackled two geometric series. These series use the formula: \[S = \frac{a}{1-r}\]

• \(a\) is the first term of the series.
• \(r\) is the common ratio between consecutive terms.

By plugging in these values, we can easily find the sum:

• The sum for \( \sum_{n=0}^{\infty} \frac{5}{2^n} \) is calculated using \( a = 5 \) and \( r = \frac{1}{2} \), yielding a sum of 10.
• Similarly, for \( \sum_{n=0}^{\infty} \frac{1}{3^n} \) with \( a = 1 \) and \( r = \frac{1}{3} \), we get a sum of \( \frac{3}{2} \).

Both of these were summed up to give us a final result, illustrating how infinite concepts can be distilled into finite answers.

Geometric Progression

A geometric progression (or sequence) is a special type of sequence where each term is found by multiplying the previous term by a constant called the "common ratio." This trait gives the sequence its name and makes it predictable and easy to work with.
In a geometric series, we build on these progressions by summing the sequence. Here’s what you need to know:

• The first term in the sequence is often represented as \( a \).
• The common ratio is represented as \( r \).
• The series progresses as \( a, ar, ar^2, ar^3, \ldots \)

The simplicity of geometric progressions is that once we know \( a \) and \( r \), we can not only write any term in the sequence but can also easily determine its convergence and sum. This makes it incredibly useful in mathematical problems, like our exercise, where distinct patterns and progressive terms play crucial roles.