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Homework Statement
Consider the series
\frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \ldots.
He then proceeds to write
\lim \textnormal{inf}_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \left(\frac{2}{3}\right)^n = 0
and
\lim\textnormal{sup}_{n\to\infty}\frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{1}{2} \left(\frac{3}{2}\right)^n = +\infty.
This is in reference to the ratio test, by the way.
Homework Equations
The Attempt at a Solution
How did he arrive at these calculations? If for the lower limit we select a_{n+1} = 1/3^n and a_n = 1/2^n, then we can recover his answer. But
\left\{\frac{a_{n+1}}{a_n}\right\} = \left\{\left(\frac{2}{3}\right)^n\right\}
is a convergent sequence and the upper and lower limits must be equal, correct? How is the upper limit \infty?
On the other hand, if we select a_{n+1} = 1/2^{n+1} and a_n = 1/3^n, then we can recover his answer for the upper limit.
Isn't there some ambiguity with selecting a_{n+1} and a_n from this series?