Sum of odd numbers, greater than zero, generates a new term. How? $$\sum_{\text{odd }k} \frac{-2e^{-ikt}}{ik\pi} = \sum_{\text{odd }k>0} \frac{-2e^{-ikt}+2e^{ikt}}{ik\pi}$$
How does limiting all $k$ values to greater than zero, introduce the new term in the numerator? 
I've been staring at this for an hour and I can't for the life of me figure out how positive $k$'s allow for a new term.
Thank you in advance for the help.
 A: $$\begin{align}
\sum_{\text{odd }k} \frac{-2e^{-ikt}}{ik\pi} &= \sum_{\text{odd }k,\ k<0} \frac{-2e^{-ikt}}{ik\pi} + \sum_{\text{odd }k,\ k>0} \frac{-2e^{-ikt}}{ik\pi}\\
&= \sum_{\text{odd }k,\ k>0} \frac{-2e^{-i(-k)t}}{i(-k)\pi} + \sum_{\text{odd }k,\ k>0} \frac{-2e^{-ikt}}{ik\pi}\\
&= \sum_{\text{odd }k,\ k>0} \frac{2e^{ikt}}{ik\pi} + \sum_{\text{odd }k,\ k>0} \frac{-2e^{-ikt}}{ik\pi}\\
&= \sum_{\text{odd }k,\ k>0} \frac{-2e^{-ikt}+2e^{ikt}}{ik\pi}
\end{align}$$
A: $\sum\limits_{k \text{ odd}} a_k= $
$\sum\limits_{k \text{ odd }>0} a_k + \sum\limits_{k \text{ odd }<0} a_k=$
$\sum\limits_{k \text{ odd }>0} a_k + \sum\limits_{k \text{ odd }>0} a_{-k}=$
$\sum\limits_{k\text{ odd} > 0} (a_k + a_{-k})$
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In this case if $j = -k < 0$ and $k = |j| = - j> 0$.
Then $\frac{-2e^{-ijt}}{ij\pi}=\frac{-2e^{ikt}}{-ik\pi} =\frac{2e^{ikt}}{ik\pi}$
So $\sum\limits_{k = \pm M} \frac{-2e^{-ikt}}{ik\pi}=$
$\sum\limits_{k = M}\frac{-2e^{-ikt}}{ik\pi} + \sum\limits_{k = -M}\frac{-2e^{-ikt}}{ik\pi}$
$\sum\limits_{k = M}\frac{-2e^{-ikt}}{ik\pi} + \sum\limits_{k = M}\frac{2e^{ikt}}{ik\pi}$
$\sum\limits_{k = M}(\frac{-2e^{-ikt}}{ik\pi}+ \frac{2e^{ikt}}{ik\pi})$
