Exercise 6.8

Question

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Exercise 8: 
    Suppose $f\in\mathscr R$ on $[a,b]$ for every $b>a$ where $a$ is fixed. Define $$\int_a^{\infty} f(x)\,dx=\lim_{b\rightarrow\infty}\int_a^bf(x)\,dx$$ if this limit exists (and
    is finite). In that case, we say that the integral on the left {\it converges.} If it also converges after $f$ has been replaced by $|f|$, it is said to converge {\it absolutely.}

Assume that $f(x)\ge0$ and that $f$ decreases monotonically on $[1,\infty)$. Prove that $\int_1^{\infty} f(x)\,dx$ converges if and only if $\sum_{n=1}^{\infty} f(n)$ converges.

analambanomenos · Accepted Answer

For any positive integer $n$ define $g_1(x)=f(n)$ and $g_2(x)=f(n+1)$ for $x\in(n,n+1]$. Since $f$ decreases monotonically, $g_1(x)\ge f(x)\ge g_2(x)$ for all $x\in[1,\infty)$.
    And since $$\int_n^{n+1}g_1(x)\,dx=f(n)\quad\quad\int_n^{n+1}g_2(x)\,dx=f(n+1)$$ we get by Theorem 6.12(b), for any positive integer $N$, $$\sum_1^Nf(n)=\int_1^Ng_1(x)\,dx\le
    \int_1^Nf(x)\,dx\le\int_1^Ng_2(x)\,dx=\sum_2^{N+1}f(n).$$ Hence $\int_1^Nf(x)\,dx$ converges as $N\rightarrow\infty$ if and only if $\sum_1^Nf(n)$ converges. In that case,
    if $A=\sum_1^{\infty} f(n)$, then $\int_1^{\infty} f(x)\,dx$ lies between $A-f(1)$ and $A$.

Exercise 6.8

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Exercise 6.8

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