I have the following series: $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$$ I want to know whether this series converges or not.
My Attempt
$$\frac{\text{nth term}}{(n-1)\text{th term}}=\frac{n-1}{n}=1-\frac{1}{n}$$
As $$n\rightarrow \infty, \frac{\text{nth term}}{(n-1)\text{th term}}=1$$
But as we know that the series does not converge, so what is wrong with my argument?
Consider the series 1 + 1 + 1 + .... The ratio of two consecutive terms is 1, but the series obviously diverges. This should help you remember that the ratio test is inconclusive when the ratio is 1, at least if you can remember another example (e.g., 1 + 1/4 + 1/9 + 1/25 + ...) which has ratio 1 but converges.