How does the ratio test fail and the root test succeed here?

[psie]

TL;DR Summary

I'm studying Spivak's Calculus, chapter 23, problem 7. There he introduces the root test and he gives an example of a series for which the ratio test fails but the root test works. I struggle with verifying this.

The series that is given is $$\frac12+\frac13+\left(\frac12\right)^2+\left(\frac13\right)^2+\left(\frac12\right)^3+\left(\frac13\right)^3+\ldots.$$ Now, it's easy to see these are two separate geometric series, however, Spivak claims the ratio test fails because the ratio of successive terms does not approach a limit. I have figured out that $$a_{2n-1}=\left(\frac12\right)^{n},\quad a_{2n}=\left(\frac13\right)^{n},\quad n=1,2,\ldots.$$ For the ratio test, we should have that the fraction $\left|\frac{a_{n}}{a_{n-1}} \right|$ approaches some finite limit or diverges to infinity. In this case, if $n$ is even we have that the fraction is $\left(\frac23\right)^n\to 0$ as $n\to\infty$. If $n$ is odd, we have $\left(\frac32\right)^n\frac13\to\infty$ as $n\to\infty$. This behavior confuses me. What can we conclude from this?

The root test given in the exercise is that if $a_n\geq0$ and $\lim\limits_{n\to\infty}\sqrt[n]{a_n}=r$, then $\sum_{n=1}^\infty a_n$ converges if $r<1$ and diverges if $r>1$. Apparently this test should work, but I do not see how when the even indexed subsequence and the odd indexed subsequence seem to converge to different limits, namely $\frac13$ and $\frac12$.

 

[PeroK]

How does Spivak define the root test? The following definition uses the $\lim \sup$.

https://en.wikipedia.org/wiki/Root_test

That said, it seems more logical to me to use the convergence of both subseries. It can't be hard to prove that if both $\sum_{n = 1}^{\infty} a_n$ and $\sum_{n = 1}^{\infty} b_n$ converge, then $a_1 + b_1 + a_2 + b_2 + a_3 + b_3 \dots$ must also converge.

PS and in this case, it's not hard actually to compute the limit of the series!

 

[Pikkugnome]

The root test is stronger than the ratio test, at least if $\limsup$-version is used. It is based on this inequality: $\liminf(a_{n+1}/a_n) \le \liminf a_n^{1/n} \le \limsup a_n^{1/n} \le \limsup(a_{n+1}/a_n)$, if every $a_n$ is a positive real number.

 

[psie]

How does Spivak define the root test?

Spivak defines the root test first as I defined it above, without the $\limsup$. Then, however, he adds that we can replace the limit with $\limsup$ and calls it the "delicate root test". He probably means that if we use the root test with $\limsup$ on the series above, then we get that $\limsup\limits_{n\to\infty}|a_n|^{1/n}=\frac12$, which is less than $1$, so we have convergence.

That said, it seems more logical to me to use the convergence of both subseries. It can't be hard to prove that if both $\sum_{n = 1}^{\infty} a_n$ and $\sum_{n = 1}^{\infty} b_n$ converge, then $a_1 + b_1 + a_2 + b_2 + a_3 + b_3 \dots$ must also converge.

I agree it would be easier to use the convergence of both subseries, but I'm trying to understand what Spivak means by the ratio test failing. We have that $\left|\frac{a_{n}}{a_{n-1}} \right|$ converges to $0$ if $n$ is even and $\infty$ if $n$ is odd, as $n\to\infty$. Can we conclude from this that $\lim\limits_{n\to\infty}\left|\frac{a_{n}}{a_{n-1}} \right|$ does not equal a finite number or infinity?

 

[PeroK]

Spivak defines the root test first as I defined it above, without the $\limsup$. Then, however, he adds that we can replace the limit with $\limsup$ and calls it the "delicate root test". He probably means that if we use the root test with $\limsup$ on the series above, then we get that $\limsup\limits_{n\to\infty}|a_n|^{1/n}=\frac12$, which is less than $1$, so we have convergence.I agree it would be easier to use the convergence of both subseries, but I'm trying to understand what Spivak means by the ratio test failing. We have that $\left|\frac{a_{n}}{a_{n-1}} \right|$ converges to $0$ if $n$ is even and $\infty$ if $n$ is odd, as $n\to\infty$. Can we conclude from this that $\lim\limits_{n\to\infty}\left|\frac{a_{n}}{a_{n-1}} \right|$ does not equal a finite number or infinity?

The ratio test is inconclusive, as every other term in the sequence of ratios is $\frac 1 2(\frac 3 2)^n$.

 

[pasmith]

I agree it would be easier to use the convergence of both subseries, but I'm trying to understand what Spivak means by the ratio test failing. We have that $\left|\frac{a_{n}}{a_{n-1}} \right|$ converges to $0$ if $n$ is even and $\infty$ if $n$ is odd, as $n\to\infty$. Can we conclude from this that $\lim\limits_{n\to\infty}\left|\frac{a_{n}}{a_{n-1}} \right|$ does not equal a finite number or infinity?

The correct conclusion is that this limit does not exist.

 

[Dinda556]

I analyze whole question and I think here limit will not exist.