Let partial sums $s_{a,d,n}$ for positive integers $a,d,n$ be defined as $$s_{a,d,n}=\sum_{k=1}^{n}\frac{1}{a+(k-1)d}$$
with $$S_{a,d}=\lim_{n\to \infty}s_{a,d,n}$$
e.g. $$S_{1,1}=S=\frac 1 1+\frac12+\frac 13+...$$ and $$S_{1,2}=\frac 1 1+\frac13+\frac 15+...$$ By rearrnging terms in S, it can be shown that $${S_{1,2}\over S}=\frac12$$
- What is $S_{a,d}/S$ ? Does it make sense to take the ratio of infinite quantities? The motivation behind this question is: "if $1/1+1/2+1/3...$ is $S$ then proprtional to that what would be $S_{a,d}".$
- Is 1. equal to $$\lim_{n\to\infty}\frac{s_{a,d,n}}{s_{1,1,n}}$$
Working it out, it seems that ,
$$\lim_{n\to\infty}\frac{s_{a,d,n}}{s_{1,1,n}}=\lim_{n\to\infty}
\frac1d\frac{\psi(n+a/d)-\psi(a/d)}{\psi(n+a)-\psi(a)}$$
where $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ is digamma.
3. Is the RHS $\frac 1d$? It seems so heurestically but i am not sure how to go about proving that.