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1) Diverges. 2) Converges. 3) Converges. 4) Converges. 5) Diverges. 6) Diverges. 7) Converges. 8) Converges to 2.5. 9) Diverges

Step-by-step explanation:

In order to find the convergence or divergence of a geometric series, we have to focus on the absolute value of its radius

The theorem states that if then the series converges to

1) The radius is r = 4 which is greater than one so it Diverges.

2) The radius is and which is less than 1 so it Converges.

3) The radius is 0.5 which is less than 1 so it Converges.

4) To figure out unknown radius, we divide the 2nd term from the first, in this case we have which is less than one so it Converges.

5) for this one, since the second term is bigger than the first, if we divide them, the answer would always be bigger than 1 so it Diverges.

The geometric series are given by the formula ∑ where c is a constant.

6) in this case our which is bigger than 1 so it Diverges.

7) and which is less than one so it Converges.

to find the sum of a convergent geometric series, we use the formula

8) plug in given information

9) since and , r is greater than 1 and it Diverges.

Hopefully the work I did was right. Have a good one!