Question

Use the limit comparison test to determine whether each of the following series converges or diverges. $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+1/n}}$$

Short answer

The series diverges.

Step-by-step solution

Identify the General Term of Series

The series we need to examine is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+\frac{1}{n}}}$. The general term of this series is $a_n=\frac{1}{n^{1+\frac{1}{n}}}$.

Simplify the General Term for Limit Comparison

Re-write the exponent in $a_n$: ${}^{1+\frac{1}{n}}=n\cdot n^{\frac{1}{n}}$. This implies $a_n=\frac{1}{n\cdot n^{\frac{1}{n}}}=\frac{1}{n}\cdot \frac{1}{n^{\frac{1}{n}}}$. For large $n$, $n^{\frac{1}{n}}$ approaches 1, so $a_n\approx \frac{1}{n}$.

Choose a Comparison Series

Choose the harmonic series $b_n=\frac{1}{n}$, which is a well-known divergent series. The goal is to use it for comparison.

Compute the Limit Comparison Criterion

Find the limit $L=\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^{1+\frac{1}{n}}}}{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n}{n^{1+\frac{1}{n}}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\frac{1}{n}}}$.

Evaluate the Limit

Evaluate $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\frac{1}{n}}}$. Since $n^{\frac{1}{n}}\approx 1$ for large $n$, the expression simplifies to $\underset{n\to \mathrm{\infty }}{lim}1=1$. Thus, $L=1$.

Apply the Limit Comparison Test Conclusion

Since $L=1(>0)$ and $b_n=\frac{1}{n}$ diverges, the Limit Comparison Test states that $a_n=\frac{1}{n^{1+\frac{1}{n}}}$ also diverges.

Key concepts

Series Convergence

In mathematics, understanding whether a series converges or diverges is crucial for various analyses. A series, which is simply a sum of terms derived from a sequence, converges if the sum approaches a certain finite number as you consider more and more terms. This concept is essential when working with infinite series, where terms are continuously added.

There are several tests to determine if a series converges, such as comparing it to other known series. By applying these tests, we can verify the nature of different series without needing to sum all terms explicitly.

The Limit Comparison Test is one such tool used to analyze series convergence by comparing a series with another known series. This test is particularly useful when the terms of the series are complex and hard to evaluate directly. If a series behaves similarly to a known convergent or divergent series, you can deduce it behaves the same regarding convergence.

Harmonic Series

The harmonic series is a fundamental example in the study of series in mathematics. It is defined as the sum $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$. Interestingly, even though its terms become very small as $n$ gets larger, the harmonic series does not converge; it diverges.

This characteristic makes it a great reference point when using the Limit Comparison Test. It helps us understand the behavior of other series with similar terms and allows us to establish conclusions about their convergence or divergence. Since the harmonic series diverges, any series compared to it, through the Limit Comparison Test with a finite, positive limit, will also diverge.

In the original exercise, by comparing the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+\frac{1}{n}}}$ with the harmonic series, we determined it diverges as well. This underlines how powerful the concept of the harmonic series is in determining the convergence nature of other series.

Divergent Series

Understanding divergent series is vital for deeper insights into infinite sums. A series is said to be divergent if its partial sums do not approach any finite limit. Instead, they continue to increase or decrease indefinitely, or they oscillate.

The original exercise was about determining whether the given series is divergent using the Limit Comparison Test. By choosing the harmonic series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ as a comparison, we found that our series diverges. This is because the comparison confirmed that the original series behaves similarly to the harmonic series, which is known to diverge.

Divergent series have significant implications in different areas of study, such as physics and engineering, where understanding the behavior of these series can inform modeling and predictions. Recognizing divergence allows us to adjust calculations and expectations when analyzing complex systems.