How to change the order of summation?

I have stumbled upon, multiple times, on cases where I need to change
the order of summation (usally of finite sums).

One problem I saw was simple

$$\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}f(i,j)=\sum_{j=1}^{\infty}\sum_{i=1}^{j}f(i,j)$$

and I can go from the first sum to the second by noting that the

constraints are

$$1\leq i\leq j<\infty$$
so the first double sum does not constrain on $i$ and constrains
$j$ to $j\geq i$. The second double summation doesn't put any constrains
on $j$ but constrains $i$ relative to $j$ $(1\leq i\leq j)$.

While this approach works for simple examples such as this. I am having
problems using it where the bounds are more complicated.

The current problem interchanges the following

$$\sum_{i=1}^{n-1}\sum_{k=2}^{n-i+1}\to\sum_{k=2}^{n}\sum_{i=1}^{n+1-k}$$

I started by writing

$$k\leq n-i+1$$

and got

$$i\leq n-k+1$$

but all other bounds are not clear to me..

the problem is that I can't use this technique since I can't write
the inequalities in the same form of

$$1\leq i\leq f(j)\leq n$$

where $n$ is some bound (possibly $\infty$).

My question is how to approach the second example by a technique that
should be able to handle similar cases