The first approach to compute DFT directly using the definition can't be correct because there is no summation over n. Try it this way. First off there really isn't a need to define a function to compute fn.
N = 715;
n = 0:(N-1);
fn = 100*exp(2i*pi*67*n/N);
Now define a function to compute the DFT for a given value of k
dftfn = @(k,fn,n,N) (sum(fn .* exp(-1i*2*pi*k*n/N))/N);
Now test for a couple of values of k
dftfn(5,fn,n,N)
dftfn(67,fn,n,N)
The next step would be to write a loop to evaluate dftfn at the values of k of interest.
Note: there are other alternatives, but I think this is the clearest.
For the second approach, I don't think the closed form expression is correct. There shoudn't be an n in any term.
fk_func = @(k,N) (100/N).*((1-exp(2i.*pi.*(67-k)))./(1-exp(2i.*pi.*(67-k)./N)));
k = 0:(N-1);
fk = fk_func(k,N);
stem(k,abs(fk)) % no need to divide by N, that's already in fk_func
As expected all the reults are tiny, but what happened at k = 67?
stem(k,abs(fk));
xlim([60 70])
We see that there is a gap, because the formula at k = 67 yields NaN because it evaluates to 0/0.
fk_func(67,N)
