SOLUTION: Find the sum of the infinite series (if it exists) <img src="http://i293.photobucket.com/albums/mm56/cool_user2004/7-1.png" alt="Photobucket - Video and Image Hosting">

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Question 172793: Find the sum of the infinite series (if it exists)

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Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Take note that the terms form a geometric sequence (since there is a common ratio and exponent involved)


Recall, the formula for a geometric sequence is a%5Bn%5D=a%2Ar%5En where n%3E=1


Since the coefficient is 300, this tells us that a=300


Since the value being raised to a power is -4%2F5, this means that r=-4%2F5


So the formula is a%5Bn%5D=300%28-4%2F5%29%5En. So, for instance, if n=3, then a%5B3%5D=300%28-4%2F5%29%5E3 (which is the third term).


Also, remember that the formula for an infinite geometric series is


sum%28a%2Ar%5Ek%2Ck=0%2Cinfinity%29=a%2F%281-r%29

Since we're starting at n=1, the series needs to be rewritten to

sum%28a%2Ar%5Ek%2Ck=1%2Cinfinity%29=a%2F%281-r%29-a



So in this case, the formula we'll use is

sum%28300%2A%28-4%2F5%29%5En%2Cn=1%2Cinfinity%29=300%2F%281-%28-4%2F5%29%29-300=300%2F%281%2B4%2F5%29-300=300%2F%289%2F5%29-300=500%2F3-300=-400%2F3



So the sum of the infinite series is -400%2F3