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All Flashcards
Explain the Alternating Series Test.
If and is decreasing, then the alternating series converges.
Why is it important that decreases in the Alternating Series Test?
It ensures that the terms are getting smaller in magnitude, allowing the partial sums to converge.
What happens if in an alternating series?
The series diverges by the Divergence Test.
Does the Alternating Series Test determine absolute convergence?
No, it only determines conditional convergence. It doesn't tell us if converges.
What is the significance of in alternating series?
is equivalent to , providing the alternating sign for the series.
State the Alternating Series Test.
If , , and is a decreasing sequence, then the alternating series converges.
What is an alternating series?
A series whose terms alternate in sign.
Define convergence in the context of series.
A series converges if the sequence of its partial sums approaches a finite limit.
Define divergence in the context of series.
A series diverges if the sequence of its partial sums does not approach a finite limit.
What is in the context of the Alternating Series Test?
is the non-alternating part of the series, i.e., the terms without the factor.