91Ó°ÊÓ

Understanding convergence criteria is crucial when analyzing an infinite series. This allows us to determine if a series approaches a finite value as we sum more and more terms.

For alternating series, which are series with terms that alternatively change sign, like our given series \( \sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+1}-\sqrt{n}) \), the Alternating Series Test provides a clear and systematic approach. It requires verifying two essential conditions:

• Each term’s absolute value, \( a_n \), must decrease and remain positive.
• The limit of \( a_n \) as \( n \) approaches infinity must be zero, \( \lim_{n \to \infty}a_n = 0 \).

In the mentioned series, our focus is on \( a_n = \sqrt{n+1} - \sqrt{n} \). First, we check if each term is positive, which it is because \( \sqrt{n+1} > \sqrt{n} \).
As \( n \to \infty \), we multiply the expression by its conjugate to make it simplified, resulting in a denominator that approaches infinity, leading \( a_n \to 0 \). These conditions confirm convergence.

When we encounter a series, we often want to know more than just if it converges; we want to know how it converges.

Absolute convergence occurs when the absolute value of a series also converges. We often check this because if a series converges absolutely, it also converges for sure, without worrying about alternating signs. For the given exercise, we examine:
\( \sum_{n=1}^{\infty} |(-1)^{n}(\sqrt{n+1} - \sqrt{n})| = \sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n}) \)

By comparing the series to \( \frac{1}{2\sqrt{n}} \), which resembles the harmonic series and diverges, we see that the corresponding positive series also diverges.

Therefore, the series does not converge absolutely. Its convergence is termed conditional, where the series converges only due to the alternating sign but not when ignoring the signs.

In the study of infinite series, understanding divergence is just as essential as understanding convergence. A series diverges if the sum of its terms does not approach any finite limit.

In the given problem, our series does not show absolute convergence because its absolute counterpart, \( \sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n}) \), behaves like the divergent harmonic series. This demonstrates divergence behavior when disregarding alternating signs.

Recognizing this divergence when checking absolute convergence is necessary because it often guides the testing process for conditional convergence. Although this aspect of divergence indicates that the positive series diverges, it simultaneously highlights that the alternating series compensates to achieve a balance leading to conditional convergence.