Question

Use the Limit Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}+100}$$

Short answer

The series diverges; compared to \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \).

Step-by-step solution

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n} + 100} \). Notice that this has the form of a fraction with a square root term added to a constant in the denominator. We suspect it's similar to a \( p \)-series, which in common form is \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Here, it suggests a comparison to \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \).

Determine the Comparison Series

We choose the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \). This is known as the \( p \)-series with \( p = \frac{1}{2} \), which is known to diverge because \( p \leq 1 \).

Apply Limit Comparison Test

Calculate the limit \( L = \lim_{n \to \infty} \frac{\frac{1}{\sqrt{n} + 100}}{\frac{1}{\sqrt{n}}} \). Simplifying the fraction gives \( \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n} + 100} = \lim_{n \to \infty} \frac{1}{1 + \frac{100}{\sqrt{n}}} = \lim_{n \to \infty} \frac{1}{1 + 0} = 1 \).

Conclusion Based on Limit Comparison Test

Since \( L = 1 \), which is positive and finite, the Limit Comparison Test states that both the original series and the comparison series have the same convergence behavior. Since \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) diverges, \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n} + 100} \) also diverges.

Key concepts

Convergent Series

A convergent series is when the sum of an infinite sequence of numbers approaches a finite limit as more and more terms are added. Imagine adding numbers forever but their sum approaches a number rather than keeping on escalating. This is what characterizes a convergent series. For a series \( \sum_{n=1}^{\infty} a_n \), if the limit \( \lim_{N \to \infty} S_N = \lim_{N \to \infty} \sum_{n=1}^{N} a_n \) exists and is finite, the series converges.

Convergence often requires the terms \( a_n \) to decrease rapidly enough to sustain this behavior. Popular tests for checking convergence include comparison tests like the Limit Comparison Test, ratio tests, and root tests. In the exercise, a comparison is drawn to see if the series in question converges or diverges through relevance to a known series.

Divergent Series

A series that does not converge is considered divergent, meaning its sum either shoots towards infinity or fails to settle on a finite value. For instance, if \( \lim_{N \to \infty} S_N = \infty \), the series diverges. Divergent series continue increasing without bound or oscillate indefinitely.

In the problem, we considered the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n} + 100} \) and determined its behavior. By comparing it to \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \), which is a divergent series since it doesn't settle to a finite number, we deduce that the original series also diverges.

P-Series

A p-series takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) and is vital in determining the behavior of infinite series. The p-value decides if the series converges or diverges. Specifically:

• The series converges if \( p > 1 \).
• The series diverges if \( p \leq 1 \).

In this exercise, the series had parts reminiscent of a p-series with \( p = \frac{1}{2} \). Given that \( 1/2 \leq 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) was used as a divergence comparison.

The p-series empowers us to quickly classify series behavior by assessing the p-value, streamlining convergence or divergence verdicts.

Mathematical Limits

In calculus, limits help explore the behavior of sequences and functions as inputs approach a specific value or infinity. A limit signifies what a sequence or function gets closer to, providing clarity about infinite processes.

The Limit Comparison Test, used in the exercise, relies on calculating a limit to determine series convergence. We determined the ratio of corresponding terms of the series \( \lim_{n \to \infty} \frac{a_n}{b_n} \) and found it to be finite and positive. Thus, it told us that both series—original and comparison—behave the same way.

Understanding limits is crucial in series tests, as they reveal the inherent correlation between diverse series and the potential outcomes as they extend towards infinity.