In mathematics, we have series, like
1, 2, 3, 4, 5, …
Should I say 'the series is convergent' or 'the series are convergent'?
Singular vs Plural – Is ‘The Series Is’ or ‘The Series Are’ Correct?
singular-vs-plural
Related Solutions
Whilst a dictionary search shows that the word is singular but plural in form, it is most commonly used (in my experience) with a verb in singular form.
Math(ematic)s is my least favourite subject.
Math(ematic)s is really hard.
This Ngram viewer helps:
EDIT: As @Araucaria points out, the "mathematics are" examples in the Ngram above are not really relevant. See his comment below.
The phrases
the same size
and
the same sizes
both have the right to exist, and are used to mean different characteristics of the objects which they describe.
If a class of objects can be dimensioned in two or more of ways, for example, pipe segments have the outer (or inner) diameter, wall thickness and the length, then, comparing two pipe segments, you can say
Those two segments have the same sizes.
if you mean that all dimensions of the two objects measure equal to each other (within some tolerance).
If in some context only specific dimension (like length, for instance) is deemed important, and other dimensions can be ignored, you can say about two pipe segments of two diameters
Those two segments have the same size.
if their length is what's currently important.
When talking of items of clothing, you can say of different pairs of shoes that they all have the same size, although no dimension of those complex shapes is the same if carefully measured: the shoe "size" is a marking, classifier, which is only approximately associated with a certain dimension.
Back to your example...
A filled sphere (a ball) has essentially only one dimension - the diameter. Often enough it is a hollow sphere, and the wall thickness can be taken as the other dimension, but if it is a solid sphere, there is no other dimension. This way when you only compare one dimension, it is more natural to use singular. But it would not be wrong to say
The diameters of those balls are [all] the same.
They have the same diameters.
Phrase "the same" when describing a dimension means "of the same value".
Best Answer
The word 'series' is both a singular and plural form. When it has the singular sense of 'one set', it takes a singular verb. When it has the plural sense of two or more sets, it takes a plural verb. For examples:
A series of lectures is scheduled.
Two series of lectures are scheduled: one for experts and one for laypeople. (The Free Dictionary).
1,2,3,4,5,....
Here we are talking about a set of numbers, so the right sentence is:
The series is convergent.