Converge and Divergence tests Flashcards
Does the sandwich theorem apply to sequences or series?
Sequences
What is the sandwich theorem?
If a sequence is always between two other sequences from a point n0, and both these sequences have the same limit, then this 3rd sequence has this same limit too.
What is the ratio test for sequences?
if the limit of |xn+1/xn| < 1 then xn is convergent and has a limit of 0.
What does the ratio test test for in sequences?
Convergence only
What does the comparison test test for in sequences? When is it used?
Divergence only, only when a sequence is bigger than one that tends to infinity, or smaller than one that tends to negative infinity
What is the meaning of absolute convergence of a series?
If the series summing |an| converges, then the series summing an is absolutely convergent.
What does it mean for the limit of an, if the series summing it is convergent?
an must have a limit of 0 for the series summing it to converge
Can a series converge if the sequence it’s summing has a limit other than 0?
No
What does the comparison test test for in series?
Convergence or divergence
What is the comparison test for convergence in series?
If |an| is less than or equal to λbn for n past n0, and the series summing bn is convergent, then the series summing an is absolutely convergent.
What is the comparison test for divergence in series?
if an is always bigger than or equal bn and the series summing bn tends to infinity then so does the series summing an.
In the ratio test, L=lim|an+1/an| is taken, what is the meaning of various values of L?
L>1: series summing an diverges
L<1: series summing an absolutely converges
L=1: series could do either
What is the alternating series test?
if an is non-increasing and its limit is 0, then (the sum of an(-1)^k) AND (the sum of an(-1)^k+1) are both convergent
What does the alternating series test test for?
Convergence only
Does the limit comparison test apply to sequences or series?
Series
