Question

(a) what is an alternating series?

(b) Under what condition does an alternating series converge?

(c) If these conditions are satisfies, what can you say about the remainder after n terms?

Short answer

Defining alternating series.

Alternating series is a series in which the terms alternate between being positive and negative.

Determining converging conditions of an alternating series.

Step-by-step solution

(a) Explaining alternating series:

An alternating series is a series whose terms are alternately positive and negative.

(b) determining the conditions for convergence of alternating series:

Convergence of alternating series:

An alternating series converge under the following conditions

(i) \(0 \le {b_{n + 1}} \le {b_n}\)

(ii) \(\mathop {\lim }\limits_{n \to \infty } {b_n} = 0\)

Finding remainder after n terms:

Remainder

\(|{R_n}| \le {b_{n + 1}}\).

Hence the answer is for convergence of an alternating series if (i)\(0 \le {b_{n + 1}} \le {b_n}\)and (ii)\(\mathop {\lim }\limits_{n \to \infty } {b_n} = 0\).

Therefore reminder is \(|{R_n}| \le {b_{n + 1}}\).