Question

Calculate the sum of the alternating series.

A) Convergent
B) Divergent
C) Oscillatory
D) Indeterminate

Answer 1

An alternating series can be convergent, divergent, oscillatory, or indeterminate depending on the behavior of the terms. In this case, with 10 units and two of them adding together to give the resultant, the series is a)convergent.

Step-by-step explanation:

An alternating series is a series in which the signs of consecutive terms are alternated. The sum of an alternating series may be convergent, divergent, oscillatory, or indeterminate.

To determine the sum of an alternating series, we look at the behavior of the terms. If the terms of the series are always positive and steadily decreasing, the series is convergent. If the terms are always positive and constant, the series is divergent. If the terms are initially positive, steadily decreasing, and become negative at the end, the series is oscillatory. If the terms are initially zero and steadily get more and more negative, the series is indeterminate.

In this case, since the series has 10 units with two of them adding together to give the resultant, it can be concluded that the series is a) convergent.

Answer 2

The sum of the alternating series is B) Divergent

Step-by-step explanation:

The alternating series' divergence indicates that the sum of the series does not converge to a finite value. It diverges, meaning it doesn't approach a specific number but rather grows indefinitely or oscillates without stabilizing.

An alternating series converges if its terms approach zero and satisfy the conditions of the Alternating Series Test. This test states that if the absolute values of the terms don't approach zero or don't meet the alternating criteria, the series diverges.

For a series to converge, the terms should decrease in magnitude and ultimately approach zero, ensuring that subsequent terms have diminishing impact. Conversely, if the terms fail to diminish or alternate irregularly without tending toward zero, the series diverges.

In the case of a divergent alternating series, the terms do not approach zero or alternate in a manner that leads to convergence. Consequently, the series does not possess a finite sum and diverges, precluding the possibility of calculating a final value.

The divergence of an alternating series emphasizes the importance of analyzing the behavior of its individual terms to ascertain convergence or divergence, as the alternating series test heavily relies on the behavior of the series' elements.