# Unconditional Convergence

#### chiph588@

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Is there a series that's unconditionally convergent but not absolutely convergent?

#### Drexel28

MHF Hall of Honor
Is there a series that's unconditionally convergent but not absolutely convergent?
Isn't that the defintion. A series is conditionally if it's convergent but it's modulus isn't. So, if it's unconditionally convergent either (I would assume) that it doesn't converge at all or it's absolutely convergent.

• chiph588@

#### chiph588@

MHF Hall of Honor
Isn't that the defintion. A series is conditionally if it's convergent but it's modulus isn't. So, if it's unconditionally convergent either (I would assume) that it doesn't converge at all or it's absolutely convergent.
That's what I thought, but I've only seen that absolutely convergent implies unconditional convergence.

#### chiph588@

MHF Hall of Honor
Wait a minute: "Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general".

This is from Wikipiedia.

I'm really curious to see a series with this property.

#### Drexel28

MHF Hall of Honor
• chiph588@

#### kompik

Is there a series that's unconditionally convergent but not absolutely convergent?
The two properties are equivalent in R and in finitely-dimensional spaces.

Try $$\displaystyle X=\ell_2$$ and the series $$\displaystyle x_n=\frac 1{\sqrt{n}}e_n$$, where $$\displaystyle e_n$$ denotes the sequence which has one on n'th place and all other terms are zeros.

If I remember correctly, this is precisely the most basic example given in the book