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$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI in Petrov's book; the original source of this theorem is the paper by Osipov.

$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}), \tag{10}\label{10}$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

Here is the table of the actual values of the relative errors $$e_{n,k}:=\frac{f_{n,k}}{2^n\Phi(z_{n,k})}-1$$ of the approximation $2^n\Phi(z_{n,k})$ of $f_{n,k}$ provided by formula \eqref{10} for $n=100$ and $k=30,40,50,60,70$:

Here is also the graph $\{(k,e_{n,k})\colon30\le k\le70\}$ for $n=100$:

Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI in Petrov's book; the original source of this theorem is the paper by Osipov.

$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI.

$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI in Petrov's book; the original source of this theorem is the paper by Osipov.

$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$

In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.

Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI.