Why does this series diverge? $\sum_{n=1}^\infty \frac{n-1}{4n-1}$

Question

So taking my original problem: $\sum_{n=1}^\infty \frac{n-1}{4n-1}$

I treated it like a limit problem as took the sum to be $\frac{1}{4}$ and since that is $<1$ for this geometric series, I assumed it converges. But it diverges and I don't really understand why.

We just started learning series this week and I'm having a little trouble catching on to the various reasons why some things converge and diverge.

Answer 1

It fails the $n$-th term test, since

$$\lim_{n \to \infty} \frac{n - 1}{4n - 1} = \frac{1}{4} \ne 0$$

Morally, this series just keeps adding up $\frac{1}{4}$ over and over again, ad infinitum - so it cannot converge.

So in this case, the limit of the sequence exists, while the limit of the sequence of partial sums does not.