Problem 62
Question
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{100} $$
Step-by-Step Solution
Verified Answer
The series diverges as the common ratio (\( |r| \)) is greater than 1.
1Step 1: Identify the First Term and the Common Ratio
The formula for our series is analogous to a geometric series. This implies that we can associate \( a \) as \( \frac{2^{1}}{100} = 0.02 \) being the first term and \( r \) as 2 being the common ratio.
2Step 2: Check the Value of the Common Ratio
In a geometric series, the series converges when \( |r| < 1 \) and diverges when \( |r| > 1 \). Here, our common ratio (\( r = 2 \)), so \( |2| > 1 \).
3Step 3: Final Verdict on Convergence/Divergence
Since \( |r| > 1 \), this means that the given series diverges.
Key Concepts
Convergence and Divergence of SeriesCommon RatioGeometric Series
Convergence and Divergence of Series
When studying the convergence and divergence of series, we're essentially asking whether the sum of an infinite sequence of numbers settles on a particular value (converges) or not (diverges). This concept is fundamental to sequences and series in mathematics.
For a series to converge, its terms must approach zero as their position in the sequence increases; otherwise, the overall sum may increase without bound or oscillate without settling on a fixed sum. The geometric series provides a clear criterion for convergence—the absolute value of the common ratio must be less than one. If the absolute value of the common ratio is greater than one, as in the given exercise, the terms of the series grow exponentially, and the series is deemed to diverge. This clear-cut rule helps identify the behavior of the series quickly.
For a series to converge, its terms must approach zero as their position in the sequence increases; otherwise, the overall sum may increase without bound or oscillate without settling on a fixed sum. The geometric series provides a clear criterion for convergence—the absolute value of the common ratio must be less than one. If the absolute value of the common ratio is greater than one, as in the given exercise, the terms of the series grow exponentially, and the series is deemed to diverge. This clear-cut rule helps identify the behavior of the series quickly.
Common Ratio
The common ratio, denoted as 'r', is a key characteristic of a geometric series. It is the factor by which we multiply each term to get the next term in the sequence. In other words, it is the ratio of a term to the immediate previous term.
The magnitude of this ratio directly influences the behavior of the series—a common ratio with an absolute value less than one causes the terms to decrease in magnitude, enabling convergence. Conversely, a common ratio greater than one results in the terms increasing and the series diverging. In our current exercise, the common ratio is identifiably 2, suggesting that each subsequent term after the first is twice as large as the term before it—which signals divergence.
The magnitude of this ratio directly influences the behavior of the series—a common ratio with an absolute value less than one causes the terms to decrease in magnitude, enabling convergence. Conversely, a common ratio greater than one results in the terms increasing and the series diverging. In our current exercise, the common ratio is identifiably 2, suggesting that each subsequent term after the first is twice as large as the term before it—which signals divergence.
Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric series is \( a + ar + ar^2 + ar^3 + ... \) for the first term 'a' and the common ratio 'r'.
When \( |r| < 1 \) the series has a finite sum, which is derived from the formula \( S = \frac{a}{1 - r} \) where 'S' is the sum of the series. This formula provides a crisp consolidation of the infinite terms to a single, finite value. The series in the exercise, however, with a common ratio of 2, does not meet this criterion and hence does not have a finite sum, categorizing it as a divergent geometric series.
When \( |r| < 1 \) the series has a finite sum, which is derived from the formula \( S = \frac{a}{1 - r} \) where 'S' is the sum of the series. This formula provides a crisp consolidation of the infinite terms to a single, finite value. The series in the exercise, however, with a common ratio of 2, does not meet this criterion and hence does not have a finite sum, categorizing it as a divergent geometric series.
Other exercises in this chapter
Problem 61
State the guidelines for finding a Taylor series.
View solution Problem 61
Verify that the Ratio Test is inconclusive for the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{1}{n^{4}} $$
View solution Problem 62
Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(1, x, \frac{x^{2}}{2}, \frac{x^{3}}{6}, \frac{x^{4}}{24},
View solution Problem 62
Bessel Function The Bessel function of order 0 is \(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\) (a) Show that the series converges
View solution