- #1
alexmahone
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Suppose $f(x)$ is continuous and decreasing on $[0, \infty]$, and $f(n)\to 0$. Define $\{a_n\}$ by
$a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$
(a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition.
(b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.
$a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$
(a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition.
(b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.