Problem 6
Question
Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^{2n}}{(1 + n)^{3n}} \)
Step-by-Step Solution
Verified Answer
The series converges by the Ratio Test.
1Step 1: Identify the Series Type
The given series is \( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \). This is a sequence raised to the power of \( n \), making it a candidate for the Ratio Test to determine convergence or divergence.
2Step 2: Apply the Ratio Test
The Ratio Test states that for \( \sum a_n \), if \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), the series converges.Let's find \( \frac{a_{n+1}}{a_n} \):\[ a_n = \frac{n^{2n}}{(1+n)^{3n}}, \quad a_{n+1} = \frac{(n+1)^{2(n+1)}}{(2+n)^{3(n+1)}} \]Then:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^{2n+2}}{(2+n)^{3n+3}} \cdot \frac{(1+n)^{3n}}{n^{2n}} \]Simplify: \[ = \left( \frac{(n+1)^{2}}{(2+n)^{3}} \right)^n \cdot \frac{(n+1)^2}{n^2} \]
3Step 3: Simplify and Take the Limit
Simplify the expression:\[ \left( \frac{(n+1)^{2}}{(2+n)^{3}} \right)^n \cdot \frac{n(n+2)+1}{n^2(n+1)} \]For large \( n \), \[ \left( \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} \right)^n \to e \]So the expression approaches:\[ \lim_{n \to \infty} \left( \frac{1}{1} \right)^n \cdot \frac{1}{1} = 0 \]
4Step 4: Conclusion Based on the Ratio Test
The limit \( L = 0 < 1 \) indicates that by the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \) converges.
Key Concepts
Ratio TestInfinite SeriesSequence Analysis
Ratio Test
The Ratio Test is a popular method used to determine whether an infinite series converges or diverges. It's particularly useful for series where the general term involves factorials, exponential forms, or powers. Here's how it works:
If you have a series represented as \( \sum a_n \), the Ratio Test requires you to look at the absolute value of the ratio of successive terms. Specifically, calculate the limit
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
The behavior of the series depends on this limit \( L \):
The Ratio Test is particularly effective when the terms of the series grow rapidly, as in the series given by the problem: \( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \). This series, because of its composition with powers of \( n \), is an ideal candidate for utilizing the Ratio Test.
If you have a series represented as \( \sum a_n \), the Ratio Test requires you to look at the absolute value of the ratio of successive terms. Specifically, calculate the limit
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
The behavior of the series depends on this limit \( L \):
- If \( L < 1 \): the series converges absolutely.
- If \( L > 1 \) or \( L = \infty \): the series diverges.
- If \( L = 1 \): the test is inconclusive, and you might need a different test.
The Ratio Test is particularly effective when the terms of the series grow rapidly, as in the series given by the problem: \( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \). This series, because of its composition with powers of \( n \), is an ideal candidate for utilizing the Ratio Test.
Infinite Series
An infinite series is essentially a sum of infinitely many terms of a sequence. Understanding whether this sum approaches a finite number (converges) or does not settle on any number (diverges) is crucial in mathematics.
To test for convergence, there are several methods available. Depending on the nature of the series, different tests could be more suitable.
For instance, the series given in our problem is:
\( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \).
The terms in an infinite series can often exhibit a specific pattern, and recognizing these patterns through tests like the Ratio Test helps determine their behavior.
In the context of this infinite series, applying the right tool or test helps us understand its ultimate behavior: whether or not it sums to a limit.
To test for convergence, there are several methods available. Depending on the nature of the series, different tests could be more suitable.
For instance, the series given in our problem is:
\( \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+n)^{3n}} \).
The terms in an infinite series can often exhibit a specific pattern, and recognizing these patterns through tests like the Ratio Test helps determine their behavior.
- Geometric series: Lack of changing powers can dictate simple convergence/divergence checks.
- Alternating series: Terms switch signs, affecting convergence behavior differently.
- Power series: Convergence depends on values inside their radius.
In the context of this infinite series, applying the right tool or test helps us understand its ultimate behavior: whether or not it sums to a limit.
Sequence Analysis
Sequence analysis is about understanding the intrinsic properties of sequences that form the terms of an infinite series. This analysis is crucial as series convergence directly depends on the behavior of these sequences.
When analyzing a sequence, often denoted as \( a_n \), \ we need to observe how it behaves as \( n \) approaches infinity. For example, the sequence in our original problem, \( \frac{n^{2n}}{(1+n)^{3n}} \), grows more complex as \( n \) increases.
Key points for sequence analysis include:
In our exercise, simplifying the term and transforming it into a form involving \( n \) helps reveal how the terms behave. The Ratio Test itself simplifies to consider this sequence behavior by examining \( \frac{a_{n+1}}{a_n} \), simplifying complexities, and making the overall analysis more manageable. With appropriate substitution and limit calculation, it helps us reach conclusions about the series convergence.
When analyzing a sequence, often denoted as \( a_n \), \ we need to observe how it behaves as \( n \) approaches infinity. For example, the sequence in our original problem, \( \frac{n^{2n}}{(1+n)^{3n}} \), grows more complex as \( n \) increases.
Key points for sequence analysis include:
- Rate of growth: understanding how rapidly the terms grow or shrink.
- Pattern observation: identifying whether the sequence behaves regularly or not.
- Comparison: relating the sequence to known basic sequences for easier analysis.
In our exercise, simplifying the term and transforming it into a form involving \( n \) helps reveal how the terms behave. The Ratio Test itself simplifies to consider this sequence behavior by examining \( \frac{a_{n+1}}{a_n} \), simplifying complexities, and making the overall analysis more manageable. With appropriate substitution and limit calculation, it helps us reach conclusions about the series convergence.
Other exercises in this chapter
Problem 6
Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {4}{2x + 3} \)
View solution Problem 6
Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {( - 1)^n x^n}{n^2} \)
View solution Problem 6
Determine whether the series is absolutely convergent or conditionally convergent. \( \displaystyle \sum_{n = 1}^{\infty} ( - 1)^{n-1} \frac {n}{n^2 + 4} \)
View solution Problem 6
Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n - 1}{n^3 + 1} \)
View solution