SOLUTION: Which of the following series is divergent? 1 + 3(1/4) + 9(1/4)^2 + 27(1/4)^3 +... 1 + 3(1/5) + 9(1/5)^2 + 27(1/5)^3 +... 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +...

Algebra ->  Sequences-and-series -> SOLUTION: Which of the following series is divergent? 1 + 3(1/4) + 9(1/4)^2 + 27(1/4)^3 +... 1 + 3(1/5) + 9(1/5)^2 + 27(1/5)^3 +... 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +...       Log On


   

Question 236799: Which of the following series is divergent?
1 + 3(1/4) + 9(1/4)^2 + 27(1/4)^3 +...
1 + 3(1/5) + 9(1/5)^2 + 27(1/5)^3 +...
1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +...
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started.

# 1


Take note that


which is a geometric series generated by the sequence for . In this case, a=1 and r=3%2F4

Now recall that an infinite geometric series only converges if abs%28r%29%3C1. Since abs%28r%29=abs%283%2F4%29=3%2F4=0.75%3C1 holds, this means that this infinite geometric series converges.

In other words, adds up to some finite number.


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# 2


Take note that


which is a geometric series generated by the sequence for . In this case, a=1 and r=3%2F5

Now recall that an infinite geometric series only converges if abs%28r%29%3C1. Since abs%28r%29=abs%283%2F5%29=3%2F5=0.6%3C1 holds, this means that this infinite geometric series converges.

In other words, adds up to some constant.




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# 3


Take note that


which is a geometric series generated by the sequence for . In this case, a=1 and r=3%2F2

Now recall that an infinite geometric series only converges if abs%28r%29%3C1. Since abs%28r%29=abs%283%2F2%29=3%2F2=1.5%3C1 is NOT true, this means that this infinite geometric does NOT converge. So the series diverges. In other words, the infinite series does not add up to some constant number.