Question

What is an alternating sequence?

Answer 1

Hint: We know that a sequence is a function from the set of natural numbers to any other set on which the sequence is defined. Now the term says “alternating” which simply means changing but with uniformity. If we think in this direction, we can come at a conclusion that the sign of the term is the only thing that can vary in the whole sequence while preserving the properties of it, which is what an alternating sequence is.

Complete step by step answer:
An alternating sequence is one which changes its sign alternatively. This can happen in two ways. Firstly, if the sequence has a positive sign in the beginning, then the sequence will have the sign assigned as $(-1)^{n+1}$ and if the first term has negative sign then the sign of the terms will be determined by $(-1)^n$. To understand this further, we see some examples here.
Consider the sequence $\{1,-1,1,-1,\ldots\}$. We can see here that the sign determining expression for this sequence is $(-1)^{n+1}$ where $n\in \mathbb{N}$. This is an example of an alternating sequence. Another example can be $\left(-\dfrac{1}{2}\right)^n$. We can see that sign changes here alternatively.

Note: The study of alternating sequences is important because comparative to normal sequences, the process of finding the limit of an alternating sequence is easier.