I am self studying a research paper in number theory I am unable to think how to prove two identities involving small o notation.
Unfortunately, I am unable to think about the way to derive them
Assume r= r(a) as an integer
Then , we have log(r) +$\frac{a-r}{a+1} log(2)$ = (1+o(1) ) log(a)
and 1+log(2)+$\frac{2r+1}{a+1} log(r+1) $ = 1+ log(2) + o(1).
Can someone please help in deriving one of the identities. (These is exactly the last page of the paper) .
Expanding the second in a power series in $a$ centered at $\infty$ (or equivalently, replacing $a$ with $1/b$ and expanding in a power series in $b$ centered at $0$), $$ 1 + \ln 2 + \frac{2r+1}{a+1}\ln(r+1) = 1 + \ln 2 + \frac{(2r+1)\ln(r+1)}{a} + \frac{-(2r+1)\ln(r+1)}{a^2} + \cdots $$ (This is literally an application of the geometric series formula $$ \frac{c}{1-r} = c + cr + cr^2, \quad |r|<1 \text{,} $$
with $r = 1/a$ and the constant term $0$ because in the limit as $a \rightarrow \infty$, $\frac{c}{a+1} \rightarrow 0$.)
Then every term on the right is decreasing in $a$, so the sum is decreasing in $a$ (i.e., is bounded by a constant on any interval $[N,\infty)$). Therefore, the right hand side is $1 + \ln 2 + o(1)$.