Does the fact that $\lim\limits_{n\to\infty} |f(n)/f(n-1)| = 1$ imply that $\lim\limits_{n\to\infty} f(n)$ exists?

Question

This question was inspired by another question I asked on this site. For that question, I had thought that if the differences in subsequent evaluations of a function, $f(n)$, defined with a domain of only the natural numbers (including $0$) converged to $0$ as $n$ approached infinity, then this would show that $f(n)$ would converge to some value. I was wrong in this assumption. Another idea I had is that if the ratio of one evaluation to the previous converged to $1$, this would also imply existence of a limit. This seems more likely because of the ratio test, but that applies only to terms of an infinite sum and only implies convergence when $|a_n / a_{n-1}| < 1$. Is my assumption true? Can it be proven somehow by the ratio test? If not, what is a counterexample?

Answer 1

Let consider as counterexample $f(n)$ such that

• $f(n)=1$ for $n$ even
• $f(n)=-1$ for $n$ odd

Answer 2

As soon as I wrote this question, I believe I found a counterexample for it. Consider the function as partial sums of the harmonic series. $$f(n) = \sum_{k=1}^n \frac{1}{k}$$ If we consider the limit of the ratio, we get $$\lim_{n\to\infty} \left|\frac{f(n)}{f(n-1)}\right| = \lim_{n\to\infty} \frac{\sum_{k=1}^n \frac{1}{k}}{\sum_{k=1}^{n-1} \frac{1}{k}} = \lim_{n\to\infty} \frac{\frac{1}{n} + \sum_{k=1}^{n-1} \frac{1}{k}}{\sum_{k=1}^{n-1} \frac{1}{k}}$$ Separating this into two fractions gives us $$\lim_{n\to\infty} \left(1 + \frac{1}{n\sum_{k=1}^{n-1} \frac{1}{k}}\right)$$ Because the harmonic series diverges, the denominator of the second fraction approaches infinity. Even if we did not know that the series diverged, we know it is greater than zero. That multiplied by $n$ as $n$ approaches infinity would show the denominator to tend to infinity anyway. This leads us to the limit being equal to $1$. Because we already knew that $\lim\limits_{n\to\infty} f(n)$ does not exist, this leads to a contradiction and disproves the original statement.