I am learning analysis on my own and I am puzzled with the following question.
Consider the series $$\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots$$ Indicate whether this series converges or diverges.
The following is what I was able to get.
First I noticed that $$a_n = \frac{1}{2^n}+\frac{1}{3^n}$$ would be a simple and nice form of this sequence, so according to the ratio test, $$\lim_{n \to \infty} \frac{1}{6} \left( \frac{3+2(\frac{2}{3})^n}{1+(\frac{2}{3})^n}\right) = \frac{1}{2}$$ therefore the series must converge.
However, I later understood the concept of upper an lower limits a little, so taking $$ a_n =
\begin{cases}
\frac{1}{2^n}, & \text{if $n$ is odd} \\
\frac{1}{3^{n-1}}, & \text{if $n$ is even} \\
\end{cases}$$ I figured out the following $$\lim_{n \to \infty} \frac{1}{2}\left(\frac{2}{3}\right)^n = 0$$ when $n$ is odd and $$\lim_{n \to \infty} \frac{1}{2}\left(\frac{3}{2}\right)^n = +\infty$$ when $n$ is even.
Through this experience I realized and really understood how a subsequence of a sequence may have multiple limits, and the supremum and the infimum of those limits give us the upper and lower limits, respectively. But the book does something different than I did, and I would like to have the following questions answered.
1),$\quad$ From my understanding, the ratio test tells us that $\Sigma a_n$ diverges when the upper limit is greater than 1. Clearly one of the limits goes to $+\infty$, so why cannot we conclude that the series diverge ?
2),$\quad$ I learned that by choice of the subsequence, the limit may or mayn't be different. In the case that it is different, how do we know what is the supremum and infimum? It would be clear that if they were $\pm \infty$ but in my case, why is it guaranteed that there is no other subsequence that has a negative limit ? Is there an algebraic way to guarantee them ? Or do we always have to conceptually understand all of the subsequential limits.
3),$\quad$ The book says the ratio test is inconclusive, which turns me back to question 1), but it used the ratio test to determine that the series converges. According to my even-odd wise definition of $a_n$, the "larger limit" (I am purposefully not saying supremum because I don't know how it's guaranteed that it is the supremum) using the ratio test can be found as $$\lim_{n \to \infty} \sqrt[n]{a_n} = \lim_{n \to \infty} \sqrt[n]{\frac{1}{2^n}} = {1 \over 2}$$ But the book says $$\lim_{n \to \infty} \sqrt[n]{a_n} = \lim_{n \to \infty} \sqrt[2n]{\frac{1}{2^n}} = {1 \over \sqrt{2}}.$$ Where did the $2n$ come from ?
Can someone help me out ?
Edit:Thanks for helping me out everyone! I really appreciate it.
Best Answer
By pairing adjacent terms, you’ve actually changed the series with which you’re working: you’re working with $\sum_{n\ge 1}\left(\frac1{2^n}+\frac1{3^n}\right)=\sum_{n\ge 1}\frac{2^n+3^n}{6^n}$, the series whose $n$-th term is $\frac{2^n+3^n}{6^n}$. The ratio test works fine on this series:
$$\begin{align*} \lim_{n\to\infty}\frac{\frac{2^{n+1}+3^{n+1}}{6^{n+1}}}{\frac{2^n+3^n}{6^n}}&=\lim_{n\to\infty}\frac{2^{n+1}+3^{n+1}}{6(2^n+3^n)}\\ &=\lim_{n\to\infty}\frac{2^{n+1}}{6(2^n+3^n)}+\lim_{n\to\infty}\frac{3^{n+1}}{6(2^n+3^n)}\\ &=\lim_{n\to\infty}\frac{2^n}{3(2^n+3^n)}+\lim_{n\to\infty}\frac{3^n}{2(2^n+3^n)}\\ &=\lim_{n\to\infty}\frac1{3\left(1+\left(\frac32\right)^n\right)}+\lim_{n\to\infty}\frac1{2\left(\left(\frac23\right)^n+1\right)}\\ &=0+\frac12\\ &=\frac12\;, \end{align*}$$
which is essentially the calculation that you made. However, if you apply the ratio test to the original series, you find that if the $n$-th term is $a_n$, then
$$a_n=\begin{cases} \frac1{2^{(n+1)/2}},&\text{if }n\text{ is odd}\\\\ \frac1{3^{n/2}},&\text{if }n\text{ is even}\;, \end{cases}$$
so that
$$\begin{align*} \frac{a_{n+1}}{a_n}&=\begin{cases} \frac{1/3^{(n+1)/2}}{1/2^{(n+1)/2}},&\text{if }n\text{ is odd}\\\\ \frac{1/2^{(n+2)/2}}{1/3^{n/2}},&\text{if }n\text{ is even} \end{cases}\\\\ &=\begin{cases} \left(\frac23\right)^{(n+1)/2},&\text{if }n\text{ is odd}\\\\ \frac12\left(\frac32\right)^{n/2},&\text{if }n\text{ is even}\;. \end{cases} \end{align*}$$
Thus, $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ does not exist: the terms with odd indices are approaching $0$, but those with even indices are increasing without bound. Since the limit does not exist, the ratio test is inconclusive.
The ratio test applied to your modified series gives the correct answer because the original series is absolutely convergent; this permits you to combine the adjacent terms as you did, but since you don’t know ahead of time that the series converges, you can’t make use of the fact to show that it converges.