Question

Determine whether the following series converge or diverge. $$\sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}}$$

Short answer

Based on the Limit Comparison Test, the given series $$\sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}}$$ diverges, because the limit of the ratio of its terms to a new series $$\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}$$ is a positive finite value (c=1), and this new p-series also diverges.

Step-by-step solution

Limit Comparison Test

The Limit Comparison Test states that if two series have the same convergence or divergence properties, then the limit of their ratio as k approaches infinity will be a constant value c, where c is a positive finite value. So, we calculate the limit of the ratio of their terms: $$\lim_{k \to \infty} \frac{\sqrt{\frac{k+1}{k}}}{\frac{1}{\sqrt{k}}}$$

Simplify the expression

Simplify the expression inside the limit in order to find the limit as k approaches infinity: $$\lim_{k \to \infty} \frac{\sqrt{k+1}}{\sqrt{k}}$$ Now, divide the numerator and denominator by \(\sqrt{k}\): $$\lim_{k \to \infty} \frac{\sqrt{\frac{k+1}{k}}}{1}$$

Evaluate the limit

As k approaches infinity, the value inside the square root converges to 1: $$\lim_{k \to \infty} \sqrt{\frac{k+1}{k}} = \sqrt{1}=1$$

Determine the convergence of the new series

Since the limit is positive and finite (c = 1), we need to determine the convergence of the new series we created, which is the sum of the reciprocal of the square root of k: $$\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} = \sum_{k=1}^{\infty} k^{-\frac{1}{2}}$$ The new series is a p-series where p = 1/2. A p-series converges if p > 1, and diverges if p ≤ 1.

Conclude about the original series

Since p = 1/2 is less than or equal to 1, the p-series, and thus the new series diverge. Based on the Limit Comparison Test, the original series will also diverge, since the limit we calculated was a positive finite value, and the behavior of the series are the same. Therefore, the given series diverges: $$\sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}}$$

Key concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool for determining the convergence or divergence of a series. It is particularly useful when you want to compare a complicated series with a simpler one whose behavior is already known. The essence of this test is to establish a likening between two series by examining the limit of the ratios of their terms.

If you have two series \( \sum a_k \) and \( \sum b_k \), and you suspect they have similar behavior, you calculate the limit \( \lim_{{k \to \infty}} \frac{{a_k}}{{b_k}} \).

• If the limit \( c \) is positive and finite, then both series either converge or diverge together.
• It’s essential to choose a comparison series \( \sum b_k \) that is easy to analyze, like a p-series, for best results.

For our series \( \sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}} \), we found that the limit ratio with the series \( \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \) is 1, therefore they share the same convergence behavior, which we then analyze further.

p-Series

A p-series is a special form of mathematical series expressed as \( \sum_{k=1}^{\infty} \frac{1}{k^p} \). Understanding the convergence of a p-series critically relies on the exponent \( p \).

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

This guideline is helpful when you encounter series that can be transformed or approximated into the form of a p-series. In the case of our problem, we transformed the related series to \( \sum_{k=1}^{\infty} k^{-rac{1}{2}} \). Here, \( p = \frac{1}{2} \) which fits the criterion \( p \leq 1 \), indicating divergence. This information plays a crucial role in deciphering the behavior of the original series through the Limit Comparison Test.

Divergence

In mathematical series analysis, divergence indicates that the sum of the infinite series does not settle on a finite value. Divergence is a critical concept since not all series have a sum that converges to a number.

When we use tests like the Limit Comparison Test, we often aim to identify divergence in series where absolute convergence isn’t apparent.

• If a series is a divergent p-series (for example, when \( p \leq 1 \)), this divergence behavior can help you deduce the behavior of more complex series by comparison.
• Critical thinking in analyzing series relies on identifying these divergence patterns to inform the overall analysis.

With our original series \( \sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}} \), finding that its related p-series diverges gives a strong clue that the initial series diverges as well.

Mathematical Series Analysis

Mathematical Series Analysis involves scrutinizing series to establish their convergence or divergence. This field is fundamental in understanding complex mathematical concepts and functions which rely heavily on infinite series representations.

Analysts employ various tests and techniques, such as the Limit Comparison Test, to break down the series properties into understandable chunks. Within this analysis, recognizing the role of each component paves the way for thorough comprehension.

• Breaking down a series into recognized simpler forms, like p-series, offers insights about convergence.
• The behavior of the individual terms (e.g., as \( k \to \infty \)) often influences the overarching behavior of the series.

For our analysis, understanding that divergence occurs provides direction in predicting and verifying results of series behavior without necessarily calculating every term involved. This approach underscores why Mathematical Series Analysis is pivotal in fields that heavily employ mathematical concepts, like physics and engineering.