Question

A. What is an alternating series? An alternating series is a ______________ whose terms are _____________

b. Under what conditions does an alternating series converge?
c. If these conditions are satisfied, what can you say about the remainder after n terms?

Answer 1

A) An alternating series is a "SERIES" whose terms are "ALTERNATIVELY POSITIVE AND NEGATIVE"

B) -It's nth term converges to zero.

- Its terms are non-increasing. This means that each term is either less than or equal to it's predecessor provided we ignore the minus signs.

C) |R_n| = |S - S_n| ≤ |a_(n+1)|

Explanation:

A) An alternating series is a "SERIES" whose terms are "ALTERNATIVELY POSITIVE AND NEGATIVE"

B) The 2 conditions for an alternating series to converge are;

-It's nth term converges to zero.

- Its terms are non-increasing. This means that each term is either less than or equal to it's predecessor provided we ignore the minus signs.

C) If the conditions in B above are satisfied, then we can write for the remainder terms that;

|R_n| = |S - S_n| ≤ |a_(n+1)|

Where;

R_n is remainder after n-terms

S is sum of the first n-terms