Divergent and Convergent sums

Question

Does sum convergence or divergence indicates anything for we can determine if the sum has an explicit formula? i.e $\sum_{k=1}^n \frac 1k$ is divergent, does this mean we can/can't devise an explicit formula for this sum?

Answer 1

There is an asymptotic estimate for this sum, usually denoted by $H_n$, namely $$H_n=\ln n+\gamma+\frac1{2n}-\frac1{12n^2}+\frac1{120n^4}+O\left(\frac1{n^6}\right) $$ Up to the Landau big Oh, this is an explicit formula.

Answer 2

No, it does not mean anything. In fact, we define the harmonic numbers as follows: $$H_n=\sum_{k=1}^{n} \frac{1}{k}$$ Because it does not have a nice closed form. But some of them have, for example: $$\sum_{i=1}^{n} i=\frac{n(n+1)}{2}$$ Or $$\sum_{i=1}^{n} i^2=\frac{n(n+1)(2n+1)}{6}$$