Question

When will an infinite geometric series with –1 < r < 0 converge to a number less than the initial term? Explain your reasoning, and give an example to support your answer.

Answer 1

If r is negative, the denominator of the formula for the sum of the series is positive and greater than 1.

If the initial term is divided by a positive number greater than 1, the result is a number smaller than the initial term.

So, if the initial term is positive, then the series will converge to a number less than the initial term.

For –1 < r < 0, an example with a1 > 0, such as 1,000 – 100 + 10 – 1 + …

Explanation:

this is the sample answer

Answer 2

If r is negative, the denominator of the formula for the sum of the series is positive and greater than 1.
If the initial term is divided by a positive number greater than 1, the result is a number smaller than the initial term.
So, if the initial term is positive, then the series will converge to a number less than the initial term. For -1<r<0, an example with a1>0, such as 1000-100+10-1+...