You can't because addition is only a binary function on Natural numbers.

To go beyond that to sums involving more than two Natural numbers you need to define something else.

As it happens $(a+b)+c=a+(b+c)$ and $a+b=b+a$ for all Natural numbers. That is, addition is both associative and commutative. As a result any finite number of additions can be done in any order and the result will be the same.

This is simply not true of an infinite number of additions. Firstly you have to define what an infinite summation means, with the default definition being:

$\sum _{i=0}^{\infty }{a}_i=\underset{n\to \infty }{lim}\sum _{i=0}^n{a}_i$ where the limit exists.

Sometimes this limit exists but the limit does not exist for absolute values of the $a_i$, in which case the series is said to be conditionally convergent. In that case the Riemann Series Theorem asserts that the sequence can be re-ordered so that it sums to any Real value you care to choose. Hence such a set of numbers does not have a unique infinite sum, let alone a set of numbers that is not even conditionally convergent – such as the entire set of Natural numbers.

On the other hand you can have a different definition of summation (they are usually chosen to coincide with the default definition where the latter converges). Some such definitions produce the value $\frac{-1}{12}$ for the sequence of Natural numbers, but note that these definitions are also order-dependent – completely unlike finite summations.

Frankly infinite summations are only loosely related to binary addition, so your intuition that the sum of all the Natural numbers "should" be infinite is of limited use.