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## Main Question or Discussion Point

I'm looking for help with my conceptual understanding of part of the following:

1)

This makes perfect sense to me.

2)

Again, makes perfect sense.

3)

Here's where I'm thrown a little. I can follow the proofs in my textbook fine, and I think I see what they all suggest.

Essentially: The RATE at which the terms of a series approaches zero (assuming they do at all) is what really determines convergence/divergence -- am I understanding this right?

Thanks,

-GM-

1)

*If a series is convergent it's nth term approaches 0 as n approaches infinity*This makes perfect sense to me.

2)

*If the nth term of a series does not approach 0 as n approaches infinity, the series is divergent*Again, makes perfect sense.

3)

*A divergent series can have an nth term that approaches 0 as n approaches infinity. Thus #1 cannot be used as a test FOR convergence.*Here's where I'm thrown a little. I can follow the proofs in my textbook fine, and I think I see what they all suggest.

Essentially: The RATE at which the terms of a series approaches zero (assuming they do at all) is what really determines convergence/divergence -- am I understanding this right?

Thanks,

-GM-