Problem 47
Question
Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} $$
Step-by-Step Solution
Verified Answer
By using the ratio test, it is determined that the series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \) diverges.
1Step 1: Definition of the Ratio Test
The Ratio Test states that for a series \( \sum_{n=1}^{\infty} a_{n} \), the limit L = \( \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \), if L < 1 then the series absolutely converges, if L > 1 then series diverges, and if L = 1 the test is inconclusive.
2Step 2: Apply the Ratio Test
The Ratio Test is applied to the given series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \) by taking the ratio \( _{n+1}/a_{n} \), giving \( \frac{\frac{2^{n+1}}{(n+1)^{2}}}{\frac{2^{n}}{n^{2}}} \). This simplifies to \( \frac{2(n^{2})}{(n+1)^{2}} \) .
3Step 3: Calculate the Limit
Next find the limit as n approaches infinity, \( \lim_{n \rightarrow \infty} \frac{2(n^{2})}{(n+1)^{2}} \). This gives a limit of 2.
4Step 4: Assess the Ratio Test Result
Since the limit from step 3, which is 2, is greater than 1, the Ratio Test determines that the given series diverges.
Key Concepts
Convergence and Divergence of SeriesLimit of a SequenceInfinite Series
Convergence and Divergence of Series
In mathematics, the convergence or divergence of a series plays a crucial role in understanding the behavior of infinite sums. A series is a sum of terms of a sequence, typically expressed as \( \sum_{n=1}^{\infty} a_n \). Determining whether these series converge or diverge helps to understand whether adding infinitely many terms leads to a finite sum.
**Convergent series** are those for which the sum approaches a finite number as more and more terms are added. For instance, the series of fractions \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, meaning that as we add more terms, the total sum will approach a certain value.
**Divergent series** are those where the sum does not approach any limit. This can mean the series either tends towards infinity or just doesn't settle to a particular number. For example, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a well-known divergent series.
Tests like the **Ratio Test** help us determine if a series is convergent or divergent. This test is especially useful for dealing with series involving factorials, exponential terms, or powers.
**Convergent series** are those for which the sum approaches a finite number as more and more terms are added. For instance, the series of fractions \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, meaning that as we add more terms, the total sum will approach a certain value.
**Divergent series** are those where the sum does not approach any limit. This can mean the series either tends towards infinity or just doesn't settle to a particular number. For example, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a well-known divergent series.
Tests like the **Ratio Test** help us determine if a series is convergent or divergent. This test is especially useful for dealing with series involving factorials, exponential terms, or powers.
Limit of a Sequence
The concept of a limit is fundamental in calculus and analysis. When considering a sequence, the limit describes the behavior of the sequence as the term number goes to infinity.
**Limit of a sequence** is denoted as \( \lim_{n \rightarrow \infty} a_n \) and refers to the value that the terms of the sequence approach as \( n \) becomes very large. If the sequence approaches a definite number, then we say it converges to that limit. If it doesn't, it diverges.
For example, consider the sequence \( a_n = \frac{1}{n} \). As \( n \) becomes very large, \( a_n \) gets closer and closer to zero. Therefore, \( \lim_{n \rightarrow \infty} \frac{1}{n} = 0 \), indicating that this sequence converges to zero.
Understanding limits allows us to use the **Ratio Test** effectively. The test itself involves finding the limit of the ratio of consecutive terms in a series.
**Limit of a sequence** is denoted as \( \lim_{n \rightarrow \infty} a_n \) and refers to the value that the terms of the sequence approach as \( n \) becomes very large. If the sequence approaches a definite number, then we say it converges to that limit. If it doesn't, it diverges.
For example, consider the sequence \( a_n = \frac{1}{n} \). As \( n \) becomes very large, \( a_n \) gets closer and closer to zero. Therefore, \( \lim_{n \rightarrow \infty} \frac{1}{n} = 0 \), indicating that this sequence converges to zero.
Understanding limits allows us to use the **Ratio Test** effectively. The test itself involves finding the limit of the ratio of consecutive terms in a series.
Infinite Series
An infinite series is simply a series with an infinite number of terms. It's an extension of a finite series to endless sums and is typically represented by \( \sum_{n=1}^{\infty} a_n \). Dive deeper into how these series work:
- **Divergence**: If adding all the terms of an infinite series results in an endless sum, the series is divergent. An example is \( \sum_{n=1}^{\infty} n \), where the sum becomes infinite.
- **Convergence**: If the series results in a finite total sum, it's convergent. A familiar convergent infinite series is the geometric series \( \sum_{n=0}^{\infty} ar^n \), where \(|r| < 1\).
Infinite series can seem daunting because we're trying to make sense of an endless process. However, techniques like tests for convergence, including the **Ratio Test**, allow us to analyze and determine whether the series behaves in a manageable way. By applying these tests, we gain insight into whether infinite series end up with meaningful sums or just dissipate into infinity. Understanding and evaluating infinite series is fundamental in calculus and mathematical analysis.
- **Divergence**: If adding all the terms of an infinite series results in an endless sum, the series is divergent. An example is \( \sum_{n=1}^{\infty} n \), where the sum becomes infinite.
- **Convergence**: If the series results in a finite total sum, it's convergent. A familiar convergent infinite series is the geometric series \( \sum_{n=0}^{\infty} ar^n \), where \(|r| < 1\).
Infinite series can seem daunting because we're trying to make sense of an endless process. However, techniques like tests for convergence, including the **Ratio Test**, allow us to analyze and determine whether the series behaves in a manageable way. By applying these tests, we gain insight into whether infinite series end up with meaningful sums or just dissipate into infinity. Understanding and evaluating infinite series is fundamental in calculus and mathematical analysis.
Other exercises in this chapter
Problem 47
Use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{n !} $$
View solution Problem 47
Verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001. $$ \sum_{n=0}^{\infty} \frac{2^{n}}{n !}=e^{2} $$
View solution Problem 47
Find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. $$ \sum_{n=1}^{\infty}(-1)^{n+
View solution Problem 48
(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{9} $$
View solution