Final answer:
The sum of the series defined by a_i = 1, 2, 3, ..., i for i = 1, 2, 3, .... is undefined.
Step-by-step explanation:
The given question is asking us to find the value of a series defined by the sequence of numbers a_i = 1, 2, 3, ..., i for i = 1, 2, 3, ....
So, the series can be written as: 1, 2, 3, 4, 5, 6, ...
This is an arithmetic sequence with a common difference of 1. To find the value of the series, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, a = 1 and the series continues indefinitely, so we can use the formula for the sum of an infinite arithmetic series:
S = (a/1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio.
Substituting the values into the formula, we get:
S = (1/1 - 1)
S = (1/0)
Since division by 0 is undefined, the sum of this series is undefined.