Final answer:
A convergent series exhibits the property of dimensional consistency within its terms, necessitating a dimensionless variable for valid addition, akin to not being able to add different physical dimensions like apples and oranges.
Step-by-step explanation:
The question pertains to the properties of convergent series in mathematics. When we talk about convergent series, we're looking at an infinite sequence of numbers or terms that come closer to a certain finite value, known as the limit, as more terms are added.
A fundamental property of such series is the consistency of dimensional analysis. This means in a power series, if every term must have the same dimension for valid addition, the variable (or argument) must be dimensionless.
This can be seen through the equation [x] = L¹M¹T¹ where raising x to any power should result in the same dimensions, which is only possible if those dimensions are zero. Hence, a variable with dimension L¹M±T± (indicating length, mass, and time respectively) must be dimensionless. This principle harks back to the concept that you cannot add quantities of different dimensions (like 'apples and oranges').