This is an example taken from "Principles of Mathematical Analysis by Walter Rudin". Example 3.35, Page 67.
Consider the series : (a) $ \frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^3}\cdots $
(b) $ \frac{1}{2}+1+\frac{1}{8}+\frac{1}{4}+\frac{1}{32}+\frac{1}{16}+\frac{1}{128}+\frac{1}{64}+\cdots$
My solution: (a) Clearly the series is generated by : < $\frac{1}{2^n}+\frac{1}{3^n}$>
lim inf $_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$= lim inf $_{n \rightarrow \infty} \frac{\frac{1}{2^{n+1}}+\frac{1}{3^{n+1}}}{\frac{1}{2^n}+\frac{1}{3^n}}$
On solving, I get
lim inf $_{n \rightarrow \infty}\frac{a_{n+1}}{a_{n}}=\frac{1}{2}$.
This is not what the book suggests. Can someone please provide with a detailed solution of what the book has done. Thanks in Advance.
Hint for a). You are not supposed to group the terms two-by-two. You have to look at the ratios of of all the terms to their predecessors. For example the ratios start with $\frac 2 3, \frac 3 {2^{2}}, \frac {2^{2}} {3^{2}}, \frac {3^{2}} {2^{3}},...$. Can you find the $\lim \inf$ of this sequence?