Question

Explain how the limit Comparison Test works.

Short answer

Answer: The series ∑(1/(n^2+1)) converges.

Step-by-step solution

Sequences and Series

A sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence. In this context, we usually deal with infinite series, which have an infinite number of terms. An infinite series is represented by the sum of the terms from n=1 to infinity: ∑(a_n).

Convergence and Divergence

A series is said to be convergent if the sum of its terms approaches a finite value as the number of terms increases. Conversely, a series is divergent if the sum of its terms does not approach a finite value (either it goes to infinity or does not have a limit).

The Limit Comparison Test

The Limit Comparison Test is a method used to determine the convergence or divergence of a series by comparing it to another series. If two series have similar behaviors as n approaches infinity, they either both converge or both diverge. The test can be stated as follows: given two series ∑(a_n) and ∑(b_n) with a_n > 0 and b_n > 0, calculate the limit L as n goes to infinity: L = lim_{n->∞} (a_n / b_n). If L > 0 and is finite, then both series either converge or diverge. That is, if one of the series is known to converge (or diverge), then the other series will also converge (or diverge).

Application of the Limit Comparison Test

Let's consider an example to understand the application of the Limit Comparison Test: Given the series ∑(1/(n^2+1)) and its comparison to the known convergent series ∑(1/n^2), also known as the p-series.

Calculating the Limit

We need to find the limit L = lim_{n->∞} ((1/(n^2+1)) / (1/n^2)). First, simplify the expression within the limit: (1/(n^2+1)) / (1/n^2) = (1/n^2)/(n^2+1) = 1/((n^2+1)/n^2) Now, find the limit as n goes to infinity: L = lim_{n->∞} (1/((n^2+1)/n^2)) = lim_{n->∞} (n^2/(n^2+1)) = 1

Conclusion

Since L = 1, which is a finite positive value, and we know that the p-series ∑(1/n^2) converges, we can conclude that the given series ∑(1/(n^2+1)) also converges, by the Limit Comparison Test.

Key concepts

Convergence and Divergence

In mathematics, when we discuss series, two important concepts arise: convergence and divergence. A series is essentially the sum of terms from a sequence which can potentially have infinitely many terms. To determine whether a series is meaningful, we need to know if it converges or diverges.
A series is said to converge if the sum of its infinite terms approaches a specific, finite value as you add more terms together. If you think of each term as a step along a journey, then the journey ends at a distinct destination when the series converges. Conversely, if a series diverges, its journey doesn't settle to a specific endpoint.
For instance, the series ∑(1/n) diverges, meaning as you keep adding 1/n starting from n=1 to infinity, the sum continuously grows without ever approaching a limit. Knowing whether a series converges or diverges allows mathematicians and scientists to determine the applicability of a series to physical models, among other uses.

Infinite Series

An infinite series is an extension of the concept of a sequence. While a sequence is simply an ordered list of numbers, an infinite series is the sum of these numbers continuing endlessly. Represented as ∑(a_n), it involves adding together terms where n starts at 1 and progresses to infinity.
Infinite series are integral in calculus and analysis, acting as approximations for more complex mathematical functions. A famous example is the series expansion for the exponential function, which can approximate functions such as \(e^x\).
Working with infinite series involves determining if they are convergent (the sum is finite) or divergent (the sum is not finite). Various tests, including the Limit Comparison Test, play a critical role in making these determinations. The properties of the series dictate how we approach solving problems and applying them in real-world scenarios.

p-series

The p-series is a specific type of series used frequently as a benchmark in calculus. It is defined as the series ∑(1/n^p), where "p" is a positive constant. Whether a p-series converges or diverges depends on the value of "p".
A crucial rule to remember is: if 𝑝 > 1, the p-series converges. Conversely, if 𝑝 ≤ 1, the series diverges. This provides a clear criterion for understanding behavior trends in series similar in form to p-series.
To illustrate, the series ∑(1/n^2) is a convergent p-series with 𝑝 = 2. Thus, the convergence properties of p-series help in applying the Limit Comparison Test. By comparing an unknown series to a known p-series, such insights can lead us to conclusions about the convergence of the more complex series, as seen in the Limit Comparison Test example involving the ∑(1/(n^2+1)) series.