Does a_n=(-1/2)^n sequence converge or diverge? How do you find its limit?

Does a_n=(-1/2)^n sequence converge or diverge? How do you find its limit?

1 Answer
Dec 21, 2017

Sequence converges.

Explanation:

a_n = (-1/2)^n

Let's look at a few terms of this sequence.

a_1 = -1/2

a_2 = 1/4

a_3 = -1/8

This is a geometric progresion (GP) with first term a_1 =-1/2 and common ratio (r) = -1/2

We are asked whether or not the sequence converges.

Consider, lim_(n->oo) a_n = lim_(n->oo) (-1/2)^n =0

:. the sequence will converge.

Now let's consider the sum of the infinite series sum_(n=1)^oo a_n

The sum of an infinite GP where absr<0 is given by a_1/(1-r)

Hence, in our example:

sum_(n=1)^oo a_n = (-1/2)/(1-(-1/2)

= -1/(2(3/2))

=-1/3

So, we can state that the sequence converges and the sum of the infinite sequence converges to -1/3