Write $a$ as a function of $n$ when $\sum_{i=1}^{n} (i + a)^{-1} = 1$

Question

Is there a good integral estimation technique I can apply here? Thanks!

Answer 1

$$\int_a^{a+n}\frac 1{x+1}dx<\sum_{i=1}^n\frac 1{a+i}=1<\int_a^{a+n}\frac 1{x}dx\\ \biggl[\ln(x+1)\biggr]_a^{n+a}<1<\biggl[\ln (x)\biggr]_a^{a+n}\\ \ln\frac{a+n+1}{a+1}<1<\ln \frac{a+n}a\\ \ln\left(1+\frac n{a+1}\right)<1<\ln\left(1+\frac na\right)\\ 1+\frac n{a+1}<e<1+\frac na\\ \qquad \quad a<\frac n{e-1}<a+1\\ \qquad\blacksquare$$