Problem 11
Question
Verify that the infinite series diverges. $$ \sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1} $$
Step-by-Step Solution
Verified Answer
The given series \(\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}\) diverges by the comparison test.
1Step 1: Identify the Series to Compare
As \(n\) tends to infinity, the series \(\frac{n^{2}}{n^{2}+1}\) tends to \(1\). Therefore, we compare it to the series \(\sum_{n=1}^{\infty} 1\).
2Step 2: Apply Comparison Test for Divergence
The comparison test for divergence states that if every term of the series \(a_n\) is greater than or equal to the corresponding term in a diverging series \(b_n\), then the series \(a_n\) also diverges. Here, \(a_n = \frac{n^{2}}{n^{2}+1}\) and \(b_n = 1\). Since \(\frac{n^{2}}{n^{2}+1} \geq 1\) for \(n = 1\) and following, we can conclude that our series diverges.
Key Concepts
Comparison Test for SeriesInfinite SeriesSeries Divergence Verification
Comparison Test for Series
The Comparison Test is a handy way to determine if an infinite series converges or diverges. When working with series, we often struggle to directly determine the convergence or divergence. The Comparison Test simplifies this. We pick a series with known behavior (either convergent or divergent) and compare it to the one we're examining. If you have a series where terms are all greater than or equal to a known diverging series, then your series will also diverge.
It's like having a buddy system in mathematics. Keep in mind this test is exceptionally useful when paired with series that resemble known geometric or p-series.
It's like having a buddy system in mathematics. Keep in mind this test is exceptionally useful when paired with series that resemble known geometric or p-series.
Infinite Series
An infinite series is the sum of an infinite sequence of numbers. Just like counting stars in the sky, such a task sounds intimidating. Mathematically, we attempt to determine whether the series "fits" into something finite (converges) or doesn't (diverges). Many series, especially without proper bounds, stretch out to infinity.
For example, the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) is famous for its divergence, meaning it grows infinitely large as more terms are added. Understanding the behavior of infinite series is crucial in various fields like physics, computer science, and engineering.
For example, the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) is famous for its divergence, meaning it grows infinitely large as more terms are added. Understanding the behavior of infinite series is crucial in various fields like physics, computer science, and engineering.
Series Divergence Verification
Verifying the divergence of a series can be carried out using the Comparison Test, as seen in our exercise. Let's delve into this specific problem: consider the series \(\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}\). It's not very evident at first glance whether it converges or diverges.
- First, by simplifying, \(\frac{n^{2}}{n^{2}+1}\) becomes close to 1 as \(n\) grows larger.- This means each term doesn't reduce to zero, a critical test for convergence usually.- Then, comparing to the divergent series \(\sum_{n=1}^{\infty} 1\), clear things up. Since \(\frac{n^{2}}{n^{2}+1} \geq 1\) for all \(n\), we deduce it doesn't converge to zero well.- Thus, the original series diverges. This method is a straightforward application of the comparison principle, ensuring we don't get lost in complex algebra.
- First, by simplifying, \(\frac{n^{2}}{n^{2}+1}\) becomes close to 1 as \(n\) grows larger.- This means each term doesn't reduce to zero, a critical test for convergence usually.- Then, comparing to the divergent series \(\sum_{n=1}^{\infty} 1\), clear things up. Since \(\frac{n^{2}}{n^{2}+1} \geq 1\) for all \(n\), we deduce it doesn't converge to zero well.- Thus, the original series diverges. This method is a straightforward application of the comparison principle, ensuring we don't get lost in complex algebra.
Other exercises in this chapter
Problem 10
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n} \cos n \pi $$
View solution Problem 10
Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{4}{3 x+2}, \quad c=2 $$
View solution Problem 11
Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{\arctan n}{n^{2}+1} $$
View solution Problem 11
In Exercises \(7-28,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \
View solution