★ Must-know
For r>0, the sequence 1/n^r approaches 0 as n approaches infinity.
The sequence r^n converges precisely when -1<r<=1; its limit is 0 for -1<r<1 and 1 for r=1.
Further detail
📌 If a_n approaches L and f is continuous at L, then f(a_n) approaches f(L).
1. What mathematical object is an infinite sequence?
2. What is a sequence in mathematical terms?
3. For which values of r does the sequence r^n converge?
What is an infinite sequence in mathematics?
An infinite sequence is an ordered list of numbers indexed by integers.
Sequence limit
Sequence terms approach L as n→∞.
When does a sequence a_n converge to a limit L?
When its terms become arbitrarily close to L as n becomes large.
Monotone + bounded
Always convergent sequences.
What defines a monotonic sequence?
It is either increasing with a_n < a_(n+1) or decreasing with a_n > a_(n+1).
Geometric sum convergence
Converges if |r|<1; sum = a/(1-r).
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