SOLUTION: Determine whether the infinite geometric series converges. If it does, then find the sum (5/4)+(5/16)+(5/64)+...

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Question 1137525: Determine whether the infinite geometric series converges. If it does, then find the sum
(5/4)+(5/16)+(5/64)+...
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

We start with 5/4 as the first term
To get the next term, we multiply by 1/4
(5/4)*(1/4) = 5/16
and then multiply that term by 1/4 to get the third term
(5/16)*(1/4) = 5/64
and so on

a = 5/4 is the first term
r = 1/4 is the common ratio
Because r = 1/4 = 0.25 is between -1 and 1, this means the infinite geometric series does converge. In other words, that r value makes -1 < r < 1 true.

So we use the formula below to find the infinite sum S
S+=+a%2F%281-r%29

S+=+%285%2F4%29%2F%281-1%2F4%29

S+=+%285%2F4%29%2F%284%2F4-1%2F4%29

S+=+%285%2F4%29%2F%283%2F4%29

S+=+%285%2F4%29%2A%284%2F3%29

S+=+%285%2A4%29%2F%284%2A3%29

S+=+%285%2Ahighlight%284%29%29%2F%28highlight%284%29%2A3%29

S+=+%285%2Across%284%29%29%2F%28cross%284%29%2A3%29

S+=+5%2F3

The geometric series converges to the sum of 5/3 = 1.6667