Question

State whether the given series converges or diverges. $$\sum_{n=1}^{\infty} \sqrt{n}$$

Short answer

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

Step-by-step solution

Identify the Series

The given series is \( \sum_{n=1}^{\infty} \sqrt{n} \). This is an infinite series where each term is the square root of the positive integers, starting from 1.

Compare with a Known Series

To determine if the series converges or diverges, compare it with the p-series \( \sum_{n=1}^{\infty} n^{-p} \). A p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \). Here, \( \sqrt{n} \) is equivalent to \( n^{0.5} \), and therefore the series resembles \( \sum_{n=1}^{\infty} n^{0.5} \).

Determine the Behavior of the Series

The series \( \sum_{n=1}^{\infty} n^{0.5} \) can also be expressed as \( \sum_{n=1}^{\infty} n^{1/2} \). Since 1/2 is less than 1, compare it with the series \( \sum_{n=1}^{\infty} n^{-1/2} \), which diverges because the exponent -1/2 is \( \leq 1 \).

Conclusion

Since \( \sum_{n=1}^{\infty} n^{1/2} \) can be compared to \( \sum_{n=1}^{\infty} n^{-1/2} \), and the latter diverges, the original series \( \sum_{n=1}^{\infty} \sqrt{n} \) diverges. Thus, the series does not sum to a finite value.

Key concepts

P-Series

A p-series is a special type of infinite series with the general form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Here, "\( n \)" is the variable index for the terms of the series, and "\( p \)" is a constant that determines the rate of decay for the terms. Understanding whether a p-series converges or diverges is crucial when analyzing series in mathematics.

• If \( p > 1 \), the series converges. This means the infinite sum adds up to a finite number.
• If \( p \leq 1 \), the series diverges. This means the sum grows without bound as more terms are added.

For example, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a p-series with \( p = 1 \), which diverges. The rule of thumb for p-series provides a straightforward way to determine the behavior of many series by comparing them to this form.

Infinite Series

An infinite series is a sum of infinitely many terms. Written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes the terms of the series, an infinite series can either converge or diverge. *

• *A series converges if the sum of its infinite terms approaches a finite limit. This means that as you add more and more terms, the total approaches a specific number.
• A series diverges if it increases without bound or oscillates as more terms are added.

The convergence or divergence of a series often requires comparing it to known series with known behaviors, such as the p-series, or using tests specifically designed to handle certain types of series. Analyzing an infinite series involves understanding its pattern and behavior as defined by its terms.

Divergence

Divergence in the context of series refers to the quality of a series where its partial sums do not approach any specific value as more terms are added. Instead, the sum either becomes infinitely large or fails to settle at a particular number. A series is said to diverge if:

• Its terms persistently add up to no finite value, as is the case with series \( \sum_{n=1}^{\infty} \sqrt{n} \), where \( \sqrt{n} \) grows too slowly for the sum to stabilize.
• The infinite series resembles or is equivalent to a known divergent series, such as a p-series with \( p \leq 1 \).

In practical terms, divergence indicates that the series, no matter how many terms are computed, will not result in a sum that can be defined as a specific number. Recognizing a divergent series is essential in mathematical analysis when determining whether further calculations are worthwhile or applicable.