I am not sure that I got that right. For example when n = 2, which is the coefficient for .
Also a_1 and a_2 appear once, a_3 and a_4 appear twice, a_5 and a_6 appear three times, a_7 and a_8 appear four times, etc. I'm not quite sure how to fit that into the coefficients since the number of a_n terms for any given n increases every two terms.
So, if I am correct about , then this is true:
The sum is an alternating harmonic series, so it reaches a maximum value of 1 and a minimum value of . Its limit as is .
So, again, if is really a partial sum of the alternating harmonic series, then
So, it converges somewhere between 1/2 and 1 times the original series.