Absolute V.S. Conditional Convergence
graph LR
A{improper integrals?} --> B(YES) -->|if integrable| integration
B --> R(comparison test)
B --> |1/x^p| c(P Series Test)
A --> C(NO) -->|series| F{nth term test}--> D{an as n approaches to infinity} -->|not equal to zero| Z(DIVERGES)
S --> |1/n^p| c(P Series Test)
D-->|equal to zero| S{Which test?} --> z(Telescoping Series Test)
S --> |r^n| f(Geometric Series Test)
H --> R
S --> H(Comparable function) --> |limit as n approaches to infinity with a_n over b_n =L| p(Limit Comparsion test)
S --> v(Alternating Series Test)
S --> Q(Let sum from n=1 to infinty a_n be a series of nonzero term) -->w(Ratio Test)
Q --> X(Root Test)
Comparison Test of Convergence and Divergence
Integral Test with error
Alternating Series Test error