# A series conditionally convergent to 0 whose sequence of partial sums is positive.

Does there exist a series conditionally convergent to 0 with sequence of partial sums {sn} such that sn is positive for every n? I considered 1-1+1/2 -1/2 +1/3 -1/3 + ....This series converges conditionally to zero but its sequence of partial sums has all odd terms positive and all even terms zero. The series (1-log2) -1/2 +1/3 -1/4 +.....which converges to zero conditionally has sequence of partial sums with odd terms positive and even terms negative. The rearrangement of 1-1/2 +1/3 -1/4 .....with one positive term followed by 4 negative terms which conditionally converges to zero is also having its sequence of partial sums with four positive terms followed by a negative 5th term and so on alternately. These examples make me believe that there does not exist a conditionally convergent series with all its sequence of partial sums positive. How to prove this or how to get a counter example? Please help!

Yes, there is:$$1-\frac34+1-1+\frac14-\frac38+\frac12-\frac12+\frac18-\frac3{16}+\frac13-\frac13+\frac1{16}-\cdots,$$whose sum is $$0$$ and whose partial sums are$$1,\frac14,1+\frac14,\frac14,\frac12,\frac18,\frac12+\frac18,\frac18,\frac14,\frac1{16},\frac13+\frac1{16},\frac1{16},\ldots$$
• You are right. My bad. I've edited my answer, adding terms like $\frac1n-\frac1n$ in the middle of my series, This is a universal method: it turns any convergent series into a conditionally convergent one. – José Carlos Santos Apr 6 '20 at 7:08