- #1
GeoMike
- 67
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I'm looking for help with my conceptual understanding of part of the following:
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-
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-