SOLUTION: Find the sum of the infinite geometric series: 1 + 3/5 + 9/25 + ..., if it exists. Thanks

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Question 42555: Find the sum of the infinite geometric series: 1 + 3/5 + 9/25 + ..., if it exists.
Thanks
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the first 'n'-terms of a geometric series: a, ar, ar%5E2,ar%5E3,.........,ar%5E%28n-1%29 is given by S+=+a%2A%281-r%5En%29%2F%281-r%29 when red%280+%3C+r+%3C+1%29.

When 'n' is very large, r%5En << 1 [means r%5En is very very less than 1].
So, obviously when 'n' is infinity, r%5En << 1 so it can be neglected (means taken as 0) in comparison to 1.
Thus the summation formula for n = infinity becomes
S+=+a%2A%281-0%29%2F%281-r%29
or S+=+a%2F%281-r%29

Here, a = 1 and red%28r=3%2F5%29 so red%280+%3C+r+%3C+1%29.
Hence the formula of summation of geometric series for infinite number of terms (n -> infinity) is applicable.

Thus the summation of the given infinite geometric series is
S+=+1%2F%281-3%2F5%29
or S+=+1%2F%282%2F5%29
or S+=+5%2F2+=+2.5

Hence, the summation of the given series exists and its value is 2.5.