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## Main Question or Discussion Point

I was reading this article of wikipedia:

Conditional and absolute convergence

It says:

"An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long."

Is that a characterization of absolute convergent sequences? (Does all conditional convergent have infinite long path of partial sums)?

I'm not sure I understand this. I would like to see an example of a conditionally convergent series (of real numbers, not complex numbers if possible), and how is the length of this path related to the non-absolute convergence of the series.

In other words: Is the definition of conditional convergence equivalent to "the length of the path of partial sums diverge"? If true, any proof?

Thanks!

Conditional and absolute convergence

It says:

"An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long."

Is that a characterization of absolute convergent sequences? (Does all conditional convergent have infinite long path of partial sums)?

I'm not sure I understand this. I would like to see an example of a conditionally convergent series (of real numbers, not complex numbers if possible), and how is the length of this path related to the non-absolute convergence of the series.

In other words: Is the definition of conditional convergence equivalent to "the length of the path of partial sums diverge"? If true, any proof?

Thanks!