Summation from a real number to a complex number

Question

What would the solution of this be:

$\displaystyle \sum_{n=1}^i n$

where $i$ is the imaginary unit. or any other formula really, I'm just interested to know how would a summation work from a real number to a complex number

Answer 1

There is no standard definition for the expression $$ \sum_{n=1}^i n, $$ where $i$ is the square root of $-1$. You could choose to give it your own definition, but if you are going to do so then be sure to clearly mention what your definition of this summation is.

If you happened to encounter such a summation in some context without additional information, my best guess would be that $i$ is in fact not the square root of $-1$, and instead just some positive integer, so that $$ \sum_{n=1}^i n = 1 + 2 + 3 + \dots + i, $$ where $i$ is some positive integer.