Question

Does the following infinite series have a sum or not and does it converge or diverge?

Answer 1

Answer:

Step-by-step explanation:

Please present these numbers as a list:  7, -21, 63, -189, ....

Otherwise it appears that you are adding them up, which is not the case.

We can tell that this is a geometric series because each new number is -3 times the previous number:  -3(7) = -21, -3(-21) = 63, and so on.  Let r = -3 and a = first number = 7.

                                                                                                       a

The sum of an infinite geometric series exists if and only if |r} < 1.  Here, |-3|, or 3, so in this case the sum of the series does not exist.  That is, the series diverges.