= Solution
\b[True for every fixed $k\geq0$.] Extract $e^n$ from inside the <logarithm>:
$$
\log(e^n+n^k)-n=\log(1+n^ke^{-n}).
$$
Choose an <integer> $m>k$. The nonnegative terms of the <exponential series> give $e^n\geq n^m/m!$ for $n>0$, whence
$$
0\leq n^ke^{-n}\leq m!n^{k-m}\longrightarrow0.
$$
Since $0\leq\log(1+u)\leq u$ for $u\geq0$, the displayed expression tends to zero and is therefore $O(1)$. The constant and cutoff may depend on the fixed $k$; no uniform assertion over unbounded $k$ is being made.
Back to article page
