Difference between Fourier transform of $\left(\frac{1}{3}\right)^{-k}$ with $k\le -1$ and $\left(\frac{1}{3}\right)^{k}$ with $k\ge 1$
As Atul has pointed out, your both sequences are not the same, but they are time-reversed versions of each other:

k = np.arange(-15, 15)

s1 = (1/3.)**(-k)*(k &lt;= -1);
s2 = (1/3.)**(+k)*(k &gt;= +1)


plt.stem(k, s1, 'r')
plt.stem(k, s2, 'g')




Accordingly, what you see in your calculations is the time-reversal property of the Discrete-Time Fourier Transform:

$$x(t)=y(-t) \Leftrightarrow X(f)=Y(-f)$$

I.e. the frequency changes its sign in your calculations, as is expected from the time-reversal property.