Question

2. When the geometric series of the form a + ar + ar2 + ar3 + . . . + arn-1 + … is defined as an infinite geometric series, the infinite sum can be found by
placing restrictions on the value of r. Why is it
necessary to have the restriction on r to evaluate the infinite sum?

Answer 1

the restriction is that r must be between -1 and 1, not including -1 and 1.
It's easy to see if |r|>=1, then a + ar + ar2 + ar3 + . . . will get very big infinitely. this is what they call the series diverge and that the infinite sum does not approach some number.
but on the other hand, if |r|<1, then they say this series converges to some number.

take for example 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... this infinite sum converges to some number. In the geometric series you'll be working with it is important to find r s that you can tell whether a series will converge (infinite sum get's closer to a particular number) or diverge (infinite sum is infinite).