Your logic would mean no irrational numbers exist.
Let's say we have an oracle O that spits out digits of any irrational number of choice, so O(1) is the first digit, O(2) is the second, etc...
So we can then write the irrational number as
Number = O(1) + O(2)/10 + O(3)/100 + ....
(ok, depending on the irrational number, the denominators can start somewhere else, but that's irrelevant, this example is for irrational numbers between 1 and 10, but you can do the same logic for values between any 2 sequential powers of 10).
For any finite positive integer, the result of this partial sum is rational, since we have a finite number of decimals, and any number with a finite number of decimals is rational. So just like your sum that converges to pi, this approaches the irrational number we started to any accuracy you want. Yet when we actually take an infinite amount of terms, so an infinite amount of decimals, so in the limit to infinity of this sum, that's when it becomes irrational, just like the other series that coverge to pi.
But if your reasoning is correct that since any partial sum is rational, the infinite sum must also be rational, then no irrational numbers exist following the proof above.
Your reasoning is not much different from people not understanding that 0.9999.... is equal to 1. In the end that's also just 9/10 + 9/100 + .... And any finite number of terms of that arbitrarily close approaches 1, but isn't equal to 1. But when you actually have an infinite number of 9's, so an infinite amount of terms, it's actually equal to 1. This equality can also easily be proven by:
1/3 = 0.33333.....
0.999999..... = 0.333333.... * 3 = 1/3 * 3 = 1