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I need to prove the following:

  1. Given the harmonic series $\sum_{n=1}^∞ 1/n$, One can add '+' and '-' Signs such that the sum of the resulting series will be zero.

  2. Will this remain true for any $\sum_{n=1}^∞ a_n$ such that $\lim_{n \rightarrow \infty}a_n = 0$?

To be honest, It doesn't feel so intuitive to me. I guess that If I knew how to add the '+' and '-' signs, All I need to do is to prove that $\lim_{n \rightarrow \infty}S_n = 0$, Wheres $S_n$ Is the partial sum of the resulting series.

Ive tried to look at this: changing the signs of the harmonic series so that it converges?

And this: If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?

But did not help much.

gosu
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  • This remains true as long as the positive summands form a monotone sequence tending to $0$. Just add summands until you arrrive at a positive number, subtract summands until you arrive at a negative number and so on. This way, you can even let the sum tend to any real number you want. – Peter Oct 08 '21 at 10:26
  • Of course the sum $a_1+a_2+a_3+\cdots$ must diverge – Peter Oct 08 '21 at 10:31

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