Does this summation identity hold? Suppose we have $\sum_{n=-\infty}^{-N} f(n)$. Is this sum equal to $\sum_{n=N}^{\infty} f(-n)$?
The reason why I ask this question is because I am trying to write $\sum_{- \infty}^{\infty} \frac{(-1)^n}{(2n-1)^3}$ as a summation of form $\sum_1^{\infty} $. Am I correct in my reasoning?

Yes, you have correctly rewritten that sum.  Your sum can therefore be rewritten as
$$
\sum_{n=-\infty}^\infty \frac{(-1)^n}{(2n-1)^3} =  
-1 + 
\sum_{n=1}^\infty \frac{(-1)^n}{(2n-1)^3} +
\sum_{n=1}^\infty \frac{(-1)^{-n}}{(2(-n)-1)^3}
$$
