$\int$

Let f and g be continuous on on $[a,\infin)$ with $0\le f(x) \le g(x)$ for all $x\ge a$, then

• if $\int_a^\infin g(x) dx$ converges, then $\int_a^\infin f(x)$ $dx$ converges too.
  • if f(x) is always smaller (circumscribe by g(x)) than g(x) and the area under the curve of g(x) is a finite number, then the area under the curve of f(x) is always a finite number smaller.
• if $\int_a^\infin f(x) dx$ diverges*, then $\int_a^\infin f(x)$* $dx$ diverges too.
  • if f(x) is always smaller (circumscribe by g(x)) than g(x) and the area under the curve of f(x) is an infinity, then the area under the curve of g(x) is always a infinity that is greater.

$\sum$

• if $\sum_{n=1}^\infin b(x) dx$ converges and $0\le a_n\le b_n$, then $\sum_{n=1}^\infin a(x)$ $dx$ converges too.
• if $\sum_{n=1}^\infin a(x) dx$ diverges and $0\le a_n\le b_n$, then $\sum_{n=1}^\infin b(x)$ $dx$ converges too.