let $f$ be a fucntion defined on $R^+$, let S= $\int_a^\infty f(x)$ with $a\ge0$. I need to know if there exists a theorem that states that if : $\lim_{x\rightarrow \infty}f(x) \rightarrow c$ where $c\neq 0$ hence the integral diverges? and if not, is there a theorem that can prove $\int_a^\infty f(x)$ divergence through the limit of $f(x)$.
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If $f(x) \to c$ where $c > 0$, there is a $d$ such that $f(x) > c/2$ for $x > d$.
Then $\int_d^M f(x) dx > (M-d)(c/2) \to \infty$ as $M \to \infty$.
A similar argument holds if $c < 0$.
marty cohen
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can i know how did you arrive to the 2nd inequality i.e $\int_d^M f(x) dx \ge (M-d)(c/2) $ please – mandez Oct 31 '15 at 21:09
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f(x) > c/2 for x > d, and the length of the interval is M-d. – marty cohen Oct 31 '15 at 21:14