How to change the limits of a summation when the index $k$ is replaced by $-k$?

Question

Is what I am doing below correct, please assist.

$$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$

Is this the rule on how to "invert" the limits, and does it matter if there are imaginary numbers in the sum; or not or is it all the same with both pure real and pure imaginary summations?

Answer 1

The correct form would be $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1+kt}$$ You handled the limits correctly, but forgot about the denominator.

A reliable way to carry this out is to plug in $k=-m$ (or some other letter) everywhere, both in the formula and in the limits (so they become $-\infty< -m\le -1$, that is $1\le m <\infty$). Then you can relabel $m$ as $k$ if you wish, since the name does not matter.

Whether the formula contains imaginary numbers or not makes no difference: arithmetical rules work the same for complex and real numbers.

Answer 2

Write out a few terms... $$ \sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt} $$ becomes $$ \frac{e^{-t}}{1+t}+\frac{e^{-2t}}{1+2t}+\frac{e^{-3t}}{1+3t}+\dots = \frac{e^{-t}}{1-t}+\frac{e^{-2t}}{1-2t}+\frac{e^{-3t}}{1-3t}\dots $$ oops.