From context, I'll assume that all the a_i are positive, or at least that all the partial products a_1 a_2 ... a_n are all eventually positive.
If that's the case, then consider taking logarithms. You'd then get
which is the arithmetic mean of the logarithms of the first n terms.
Now, set b_n := log a_n. Then your partial sum is the Cesàro mean associated with the sequence (b_n). If a_n converges to a>0, then log a_n = b_n converges to log a; that is, (b_n) converges to log a. Since it's known that if (b_n) is any convergent sequence, its sequence of Cesàro means (1) converges and (2) its limit is the same as that of b_n, that would give you the desired result. If that's not a familiar result, then here's what would suffice to give you the result above:
Proposition: Let (b_n) be a convergent sequence in R such that b_n → b. Then if
c_n := (b_1 + b_2 + ... + b_n)/n,
the sequence (c_n) is convergent, and c_n → b, too.
A converse to the above fails, though. Namely, you can have a sequence whose associated sequence of Cesàro means converges, but the sequence itself diverges. (E.g., 1,-1,1,-1,1,-1,... diverges, but its sequence of Cesàro means converges to 0.)
One additional word of caution: the convergence of infinite products is a bit more delicate than that of series. Anything you can do to translate a statement about infinite products into one about infinite series instead will make it less of a headache to keep track of all the subtleties concerning convergence of infinite products.
Hope this helps. Good luck!