Substitution $t=-t$ in series

Question

Let's say I want to make a substitution $t=-t$ in a series $$\sum_{t=-\infty}^{\infty}a_t$$

The result of this substitution should be $$\sum_{t=\infty}^{t=-\infty}a_{-t}$$

correct? Based on the symmetry between $t$ and $-t$

Answer 1

I don't agree with drhab's interpretation of $\displaystyle\sum_{t=-\infty}^{\infty}a_t$ in the comments; however, I do think that this notation is problematic in the sense that it is in general not well-defined.

Recall the definition of $\ \displaystyle\sum_{n=0}^{\infty}x_n\ $ is: $\ \displaystyle \lim_{k\to\infty} \left(\sum_{n=0}^{n=k}x_n\right).\ $

The interpretation of the expression $\displaystyle\sum_{-\infty}^{\infty}x_n\ $ that tries to follow suit is: ... ? Well, there isn't one. Since we haven't specified a starting value of our series at each stage in the limit, my (the?) interpretation of this expression is:

$\displaystyle \lim_{k\to\infty} \left( \sum_{n=1}^{n=k} x_{f(n)} \right)\quad $

where $\ f:\mathbb{N}\to\mathbb{Z}\ $ could be any bijective function from $\mathbb{N}\ $ to $\ \mathbb{Z}.$

Since different bijections $f$ can give different values of the expression by the Riemann Series Theorem, it follows that the expression is not well-defined.

Only if we know in advance that the series is absolutely convergent does the order of terms not matter and therefore the expression would be well-defined.