Is summation the equivalent of a for loop? (Programming)

Question

It says here that summation is in this form. It got me thinking for those of you who know programming, isn't this basically the same as a for loop to some extent? Since i is some sort of a variable (index which is incremented by one every time) holding m whilst n is where the incrementation stops when i = n. An example of how this would be in PHP:

<?php 
for ($x = 0; $x <= 10; $x++) {
    echo "The number is: $x <br>";
} 
?>

\$x being our i in this case and 10 being n while the $x++ is the incrementation that in our case; i does until it is the same value as n.

Answer 1

Yes! Both summations and for loops are designed as a shorthand of saying "I want to do this a whole bunch of times but I don't really want to write it all out. Here's the step to repeat and the pattern each step varies by, have at it ;) ."

I could define some variable x = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100, or I could just say x = [the sum of the squared integers 1 to 10, inclusive]. As you pointed out, this is just a summation: sigma from i=1 to 10 of i^2; if you were to ask a computer scientist to add those numbers together, immediatly (s)he would type something like this:

int x = 0;
for(int i = 1; i <= 10; i++) {
    x += i*i;
}

As I'm sure you recognize, this is a rather trivial example. The power of a for loop comes not from adding a bunch of numbers but from executing statements of code with the altered variable as a parameter. Whereas sigma can be thought of as a function that sums the results of some function (as it really is, in its most general form, sigma from i=[lower bound] to [upper bound] of f(i), where f(x) is some function... sorry about the formating, I'm not sure how to do mathematic symbols on SE), a for loop can perform actions with those values of f(i).

I'm not sure if that makes sense, but I guess the TL;DR summery would be that yes, a summation is just a specific kind of for-loop, but a for-loop is much more expressive as it allows the programmer to define what to do with the values of its incrementing variable(s). I suppose you could say there's greater logical control in a for-loop than you find in a summation.

Answer 2

$\sum_{n=a}^b f(n)$

is equivalent to

<?php 
$sum = 0;
for ($n = a; $n <= b; $n++) {
  sum +=f(n);
} 
?>

Or in Python, sum([f(n) for n in range(a,b+1)])