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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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) (Show Source): You can put this solution on YOUR website!
I'll do the first one to get you started.
# 1
Take note that +9\left(\frac{1}{4}\right)^2+27\left(\frac{1}{4}\right)^3+\cdots=1+\left(\frac{3}{4}\right)^1+\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots)
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, +9\left(\frac{1}{4}\right)^2+27\left(\frac{1}{4}\right)^3+\cdots)
adds up to some finite number.
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# 2
Take note that +9\left(\frac{1}{5}\right)^2+27\left(\frac{1}{5}\right)^3+\cdots=1+\left(\frac{3}{5}\right)^1+\left(\frac{3}{5}\right)^2+\left(\frac{3}{5}\right)^3+\cdots)
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, +9\left(\frac{1}{5}\right)^2+27\left(\frac{1}{5}\right)^3+\cdots)
adds up to some constant.
--------------------------------------------
# 3
Take note that +9\left(\frac{1}{2}\right)^2+27\left(\frac{1}{2}\right)^3+\cdots=1+\left(\frac{3}{2}\right)^1+\left(\frac{3}{2}\right)^2+\left(\frac{3}{2}\right)^3+\cdots)
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.
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