Let's see if I can explain.

(A): Do you think 3 + 5 is a series or a sum?

(B): Do you think {3, 5} is a series or a sum?

And (C): Do you think 1+2-3+4+5-9+pi+10+11-21 is a series or a sum?

And what about pi itself, does that value diverge or converge?

A: Both

B: Neither.

C: Both

Pi is a number. The term convergence is generally applied to an infinite series or sequence. I think the question, though semantically correct, is meaningless. I could be wrong about this, though.

But this is largely irrelevant. We are asking whether the

*infinite* series mentioned above converges. By your own definition this series does not converge. Your definition stated:

"A series S{a(n)} is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit."

No such finite limit exists for these partial sums. Or, if it does, as N approaches infinity, can you tell me this limit. Or can you prove that one exists? You can not.

S1 = 1

S2 = 3

S3 = 0

S4 = 4

S5 = 9

S6 = 0

S7 = pi

S8 = Pi+10

S9 = pi + 21

S10 = pi

S11 = pi + 22

S12 = pi + 45

S13 = pi

S14 = pi + 46

S15 = pi + 93

S16 = pi

This sequence continues like this forever of course. As N approaches infinity there is no limit that the sequence of partial sums converges to. Hell, the sequence isn't even bounded. One of the conditions for convergence is that the sequence must be bounded.