What is an infinite sequence in mathematics?
An infinite sequence is an ordered list of numbers indexed by integers.
When does a sequence a_n converge to a limit L?
When its terms become arbitrarily close to L as n becomes large.
What defines a monotonic sequence?
It is either increasing with a_n < a_(n+1) or decreasing with a_n > a_(n+1).
What does every bounded monotonic sequence do?
It converges.
Sequence limit
Sequence terms approach L as n→∞.
Monotone + bounded
Always convergent sequences.
Geometric sum convergence
Converges if |r|<1; sum = a/(1-r).
Series divergence test
Diverges if terms don't tend to zero.
Partial sum
Sum of first n terms of series.
Power series
Infinite sums with variable x.
Taylor series
Series expansion of functions about a point.
Answer 10 questions on Infinite Series and Taylor Methods.
1. What mathematical object is an infinite sequence?
2. For which values of r does the sequence r^n converge?
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