Yes.
Why would you break the pattern? (Assuming you know what I mean.) Obviously, if you break the pattern, you might not get a number that is less than (1).
In the case of (\sum_{i=1}^{n}), (i) starts with (1), increases by (1) and ends with (n). The number of terms is (n), so the sum stops (is complete) after (n) number of terms.
In the case of (\sum_{i=1}^{\infty}), (i) starts with (1), increases by (1) and does not end. The fact that (i) does not end tells us that the number of terms is (\infty). This means the sum stops (is complete) after an infinite number of terms. (I assume you’re one of the people in this thread who have no problem with the concept of “actual or completed infinity”.)
(\sum_{i=1}^{\infty}) does not mean that (i) ends with (\infty). In other words, (\sum_{i=1}^{\infty} i) is not equal to something like (1 + 2 + 3 + \cdots + \infty). (Note that such a sum would have more than (\infty) terms.)
The value of (i) is always a natural number. So if we are asking a question such as “What’s the value of the sum of terms whose index is (i, 1 \leq i \leq x)?” then (x) cannot be anything other than a natural number since the range of (x), in such a case, must be the range of (i) – and this means that (x) can’t be (\infty). But if we’re asking a question such as “What’s the value of the sum after (x) number of terms?” then the range of (x) goes from (1) to the number of terms of the sum. If the number of terms is infinite, then (x) can be (\infty). And it is precisely this question that we’re asking.
What is the value of the sum (\sum_{i=1}^{\infty} \frac{9}{10^i}) after an infinite number of terms?
You’re saying it’s (1), I am saying it’s less than (1).
My argument (which is basically James’s argument) is that the pattern of this sum prohibits its value after (x) number of terms to be equal to (1) for any (x > 0). (You can limit the value of (x) to numbers that have no fractional component, if you want.) Since (\infty) is greater than (0), it applies to (\infty) as well.
By what logic does the value of this sum become (1) after an infinite number of terms?