Question

Use the Limit Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+n}}$$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+n}} \) diverges, compared with the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \).

Step-by-step solution

Identify the original series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+n}} \). We need to determine whether this series converges or diverges.

Choose a comparison series

For the Limit Comparison Test, find a simpler series that resembles the given series. Notice the dominant term in the denominator is \( n^2 \). So, we compare with the series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known as the harmonic series and is divergent.

Set up the Limit Comparison Test

According to the Limit Comparison Test, for \( a_n = \frac{1}{\sqrt{n^{2}+n}} \) and \( b_n = \frac{1}{n} \), compute \( \lim_{n \to \infty} \frac{a_n}{b_n} \).

Simplify the limit expression

Compute the limit: \[ \lim_{n \to \infty} \frac{\frac{1}{\sqrt{n^2+n}}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} \].

Evaluate the limit

Simplify the expression: \[ \lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+\frac{1}{n}}} = 1 \].

Conclude using the Limit Comparison Test

Since the limit \( \lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} = 1 \) is a positive finite number, the Limit Comparison Test tells us that \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+n}} \) converges or diverges whenever \( \sum_{n=1}^{\infty} \frac{1}{n} \) does. Since the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is known to diverge, the given series also diverges.

Key concepts

Convergence

Understanding whether a series converges is a fundamental part of analyzing series in mathematics. A series is said to be convergent if the sum of its terms approaches a finite number as the number of terms increases indefinitely. Essentially, if you keep adding more and more terms of the series, and the total sum settles on a specific number, then you have convergence.
In practical applications, convergence reveals stability or sustainable outcomes over time, making it a crucial concept for mathematicians and scientists alike.

To test convergence, various techniques are employed, such as the Limit Comparison Test. This involves comparing the series in question with another series whose convergence behavior is already known. If the unknown series behaves similarly to a known convergent series, it, too, is likely to converge.

Divergence

On the flip side, divergence is when the sum of a series does not approach a finite number, no matter how many terms are added. Instead, the sum might grow infinitely large or fluctuate without settling.
In the realm of mathematics and physics, divergence indicates beliefs that equations or processes do not stabilize but instead veer towards infinity or indefiniteness. Identifying divergence helps in understanding the limitations or expansive nature of certain series, indicating when a potential solution or model becomes unmanageable.

To establish divergence, tests like the Limit Comparison Test come into play. For instance, in the given problem, the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+n}} \) is shown to diverge when compared with the harmonic series, which is well-known for not converging to a finite value.

Harmonic Series

The harmonic series is one of the most famous examples when discussing convergence and divergence. It is expressed as \( \sum_{n=1}^{\infty} \frac{1}{n} \), and although its terms get smaller as \( n \) increases, the series itself is divergent.
Despite its terms shrinking, the harmonic series diverges because the sum increases without bound. This makes it an essential benchmark for comparison tests like the Limit Comparison Test since any series similar in behavior to the harmonic series is likely to diverge too.

Historical importance and simplicity make the harmonic series a common comparison target in related mathematical problems, such as the one given above.

Infinite Series

Infinite series form the backbone of much of calculus and mathematical analysis. An infinite series is simply a sum of an infinite sequence of terms. You begin with a sequence of numbers and attempt to sum them up infinitely.
These series are everywhere in mathematics, from calculating the value of \( e \) (natural logarithm base) to evaluating the behavior of functions and solving equations.

Infinite series can exhibit both convergence and divergence. Their analysis often involves techniques like the Limit Comparison Test to predict their behavior as the number of terms grows larger and larger. Understanding infinite series is critical in fields such as engineering, physics, and economics, where they model everything from waveforms to growth rates.