Infinite Sequences
Key Questions
-
An infinite sequence of numbers is an ordered list of numbers with an infinite number of numbers.
An infinite series can be thought of as the sum of an infinite sequence.
-
The sequence
#{a_n}# converges if#lim_{n to infty}a_n# exists (having a finite value); otherwise, it diverges.
I hope that this was helpful.
-
Informally, and (real-valued) infinite sequence is just an infinite list of real numbers
#x_{1},x_{2},x_{3},x_{4},\ldots# .More precisely, an infinite sequence is a function whose domain can be taken (among other things) to be the set of positive integers
#NN=\{\1,2,3,4,\ldots\}# and whose codomain is the set of real numbers#RR# . The output of the sequence at the input#n\in NN# is#x_{n}\in RR# .
Questions
Tests of Convergence / Divergence
-
Geometric Series
-
Nth Term Test for Divergence of an Infinite Series
-
Direct Comparison Test for Convergence of an Infinite Series
-
Ratio Test for Convergence of an Infinite Series
-
Integral Test for Convergence of an Infinite Series
-
Limit Comparison Test for Convergence of an Infinite Series
-
Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
-
Infinite Sequences
-
Root Test for for Convergence of an Infinite Series
-
Infinite Series
-
Strategies to Test an Infinite Series for Convergence
-
Harmonic Series
-
Indeterminate Forms and de L'hospital's Rule
-
Partial Sums of Infinite Series