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Is there exist any series $\sum_{n=0}^\infty a_n$ such that $\sum_{n=0}^\infty a_n$=0 but not absolutely convergent

I have no idea how to consturct such type of series

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up vote 4 down vote accepted

Sure, just take any known alternating series that converges but not absolutely, shift all the terms to the right 1, and add in a first term that is equal to the opposite of the original sum.

The fact that you added one term to the beginning won't change anything about whether it is convergent or not, or absolutely convergent or not. It will only change the sum to 0.

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Yes, there are. Consider $\sum_{i=1}^\infty (-1)^i/i $. You have that this converges to $-ln(2)$. Take $\sum_{i=1}^\infty (-1)^i /i + ln(2)$.

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Let $a_{2k}:=\frac 1k$, $a_{2k+1}:=-\frac 1k$. Then $\sum_{j=1}^{2n}a_j=\sum_{j=1}^{n}a_{2j}+\sum_{j=1}^{n-1}a_{2j+1}=\frac 1n$ and $\sum_{j=1}^{2n+1}a_j=0$ so $\sum_{j=0}^{+\infty}a_j=0$. But $$\sum_{j=1}^{2n+1}|a_j|=2\sum_{j=1}^n\frac 1j,$$ which proves that the series is not absolutely convergent.

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There is a theorem which says that if a series is convergent but not absolutely convergent, then by rearranging its terms you can get any real value. (That's why generally you cannot change the order of summation in a series which is not absolutely convergent.)

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