Question

Determine whether the following series converge or diverge. $$\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+1}}$$

Short answer

Answer: The given series diverges.

Step-by-step solution

Identify the given series

The given series is $$\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+1}}$$.

Simplify the given series

For large values of k, the term $$\sqrt{k^{2}+1}$$ is approximately equal to $$\sqrt{k^{2}}$$ which is equal to $$k$$. So, we can rewrite the given series as: $$\sum_{k=0}^{\infty} \frac{k}{k}$$.

Cancel terms in the simplified series

We can cancel out the k terms in the simplified series: $$\sum_{k=0}^{\infty} 1$$.

Compare with the harmonic series

Since the simplified series is simply the sum of 1 repeated infinitely, it is equivalent to the harmonic series, which is known to diverge. Therefore, the given series can be compared to the harmonic series, and both should have the same convergence properties.

Determine the convergence or divergence of the given series

Since the harmonic series is known to diverge, and the given series is comparable to (and shares the same convergence properties with) the harmonic series, we can infer that the given series also diverges.

Key concepts

Harmonic Series

The Harmonic Series is one of the most well-known series in mathematics. It's defined as the sum of terms in the form \( \sum_{n=1}^{fty} \frac{1}{n} \). This series grows without bound as more terms are added, meaning it does not converge to a limit. The terms of the Harmonic Series decrease in size, but they do not decrease fast enough for the sum to settle on a finite value. A good way to picture this is to imagine the series as a "long and slow climb" towards infinity.

• Each term \( \frac{1}{n} \) becomes smaller and smaller, but infinitely many terms still contribute to the overall sum.
• Even as \( n \) grows larger, the series keeps adding more to the total, never reaching a stable endpoint.

Understanding the behavior of the Harmonic Series is crucial for analyzing series that resemble it.

Divergence

Divergence in the context of series means that the sum does not converge to a finite number. Instead, it increases towards infinity. This happens when the terms of the series collectively keep increasing the overall sum, rather than approaching a specific limit.
In our example, the series \( \sum_{k=0}^{\infty} 1 \) is essentially an infinite sum of the constant value 1. Since it never approaches a particular value, it diverges. This means:

• The series has no finite bound.
• The sum keeps growing larger with each additional term.

Divergence is a key concept because it helps us distinguish between series that stabilize versus those that do not.

Limit Comparison Test

The Limit Comparison Test is a powerful tool when determining the convergence or divergence of a series. It involves comparing a given series to another series with known behavior. If the known series converges or diverges, we infer the same for the series in question. In the exercise, we simplified \( \sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+1}} \) to \( \sum_{k=0}^{\infty} \frac{1}{1} \), which can be compared to the Harmonic Series.
Here's how the Limit Comparison Test works:

• Choose a comparison series that has known convergence properties.
• Analyze the limit of the ratio of terms between the series as \( k \to \infty \).
• If the limit is a non-zero finite number, both series either converge or diverge together.

This test is a handy strategy when a straightforward analysis seems complex.

Infinite Series

An Infinite Series is a sum of an infinite sequence of numbers. Each term in the series is viewed as being added to an infinite sum. Convergence and divergence evaluate whether this sum settles to a finite value or grows indefinitely. These are the building blocks of mathematical analysis.
A key aspect of Infinite Series is recognizing their potential patterns and rates of decay (or growth). For example:

• Some series, like the geometric series, converge when their terms diminish at a fast enough rate.
• Others, such as the harmonic series, maintain a slow growth that does not allow for convergence.

Understanding Infinite Series allows mathematicians to solve problems involving growth, decay, and approximate large sums with precision.