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 +...

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# 1


Take note that


which is a geometric series generated by the sequence for . In this case, and

Now recall that an infinite geometric series only converges if . Since 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, and

Now recall that an infinite geometric series only converges if . Since 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, and

Now recall that an infinite geometric series only converges if . Since 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.