Problem 48 - Single Variable Calculus

Question

Determine whether the given series converges or diverges. If it converges, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{2^{n}-5^{n}}{3^{n}}\right)\)

Step-by-Step Solution

Answer

The given series \(\sum_{n=1}^{\infty}\left(\frac{2^{n}-5^{n}}{3^{n}}\right)\) can be represented as a difference of two geometric series. The first geometric series converges, while the second one diverges. Since the second geometric series diverges, the overall given series also diverges. Therefore, we cannot find the sum of the given series since its sum is not finite.

Step 1: The general term for our series is given by \(\frac{2^{n}-5^{n}}{3^{n}}\). Our goal is to determine if this series converges or diverges. #Step 2: Determine if the series is a geometric series or not#

To figure out if our series is a geometric series, we need to rewrite the general term in the form of \(ar^{n-1}\), where 'a' is the first term and 'r' is the common ratio. Let's try to rewrite the given general term: \[ \frac{2^{n}-5^{n}}{3^{n}} = \frac{1}{3^n}(2^n - 5^n). \] We can rewrite this as a difference of two geometric series: \[ \frac{1}{3^n}(2^n - 5^n) = \frac{1}{3^n} \cdot 2^n - \frac{1}{3^n} \cdot 5^n. \] This form shows that the given series is the difference of two geometric series, each with its own first term and common ratio. #Step 3: Determine if the individual geometric series converge#

Step 2: In order to find out if a geometric series converges or diverges, we need to look at its common ratio. For the first geometric series, we have \(a_1 = \frac{1}{3} \cdot 2\) and the common ratio \(r = \frac{2}{3}\). Since \(|r| 1\), the second geometric series diverges. #Step 4: Conclude the convergence or divergence of the given series#

Since the second geometric series in the difference diverges, the overall given series also diverges. Therefore, we cannot find the sum of the given series since its sum is not finite. The given series \(\sum_{n=1}^{\infty}\left(\frac{2^{n}-5^{n}}{3^{n}}\right)\) diverges.

Key Concepts

Understanding Convergence in Series

Convergence of an infinite series implies that as you keep adding more and more terms, the series approaches a fixed value, known as the series' sum. In simpler terms, the partial sums of the series eventually stabilize and do not stray off to infinity. To determine convergence, mathematicians typically track the behavior of these partial sums as the number of terms heads towards infinity.

• If the terms of a series shrink rapidly enough, the overall sum can settle to a specific value (converges).
• If not, the series will not converge, and its sum is considered infinite or non-existent.

For our series, after evaluating the individual portions as derived from the given expression, it was necessary to assess convergence properties, which revealed divergence, meaning the sum does not settle to any value.

Explaining Geometric Series

A geometric series takes the form \[ a + ar + ar^2 + ar^3 + \cdots \]Here, 'a' is the first term, and 'r' is the common ratio between terms. Drafting a geometric series is like stepping up by powers of the ratio 'r' each time. With geometric series, a special feature is that when the absolute value of 'r' is less than one (\( |r| < 1 \)) it converges. This is crucial in determining convergence.To break down our original expression, it can be interpreted as the difference between two geometric series over a common base. Each part has its ratio and behavior, determining whether it converges or diverges based on these principles.

• The first series with \( r = \frac{2}{3} \) is convergent.
• The second series with \( r = \frac{5}{3} \) diverges because its ratio is greater than one.

Concept of Divergence

Divergence occurs when an infinite series does not have a finite limit. This implies that as you add more terms, the series sum does not stabilize at any number, instead it increases (or oscillates) indefinitely.

• In the considered exercise, divergence was identified because one of the individual geometric series had a common ratio greater than one.
• This means this particular series continues growing without bounds, leading to the entire series itself diverging since part of it is unbounded.

Understanding divergence is key in knowing when calculating a series sum is infeasible as it doesn’t adhere to settling down to a sum.

Calculating Series Sum

The sum of a series, when it converges, is a valuable quantity that defines the complete addition of its infinite terms. It represents the limit or the value toward which the partial sums of the series tend as you include more terms.

• If a series is known to converge, techniques can be applied, for instance summing formulas for geometric series: \[ S = \frac{a}{1-r} \] when \( |r| < 1 \)
• This formula helps in finding the precise sum when the series overall fits the geometric mold.

However, with the given problem's outcome being divergence, it's impossible to calculate a precise finite series sum without resolution.