Question

Detailed proof please.

. 1. Determine whether the following statements are true or false. If one is true, provide a proof. If one is false, provide a counterexample (proving that it is in fact a counterexample). IF f is a positive continuous function on [1,00) and (f(x))2dx converges, THEN Sº f(x)dx converges. • IF f is a positive continuous function on [1,00) such that limx700 f(x) O and soon f(x)dx converges, THEN S ° (f (x))2dx converges. IF f is a continuous function on [1,00) and sº f(x)dx converges, THEN 2n=1 f(n) converges. • IF f is a continuous function on [1,00) and sº f(x)dx diverges, THEN [=1 f(n) diverges. =

Answer 1

• FALSE. Let
  f(0) = T
  . Then
  1 $, (5m)?dx = 1, dir - 1 C
  converges but $\int_1^\infty f(x)\,\text dx =\int_1^\infty \frac1x\,\text dx = \left[ \ln x \right ]_1^\infty =\infty$ does not.
• TRUE. Since
  lim f(0) = 0 T-
  there exists a
  R>
  such that for $x\geqslant R$ . For $x\geqslant R$ we have $(f(x))^2\leqslant f(x)$ . Then,
  $\int_1^\infty (f(x))^2 \,\text dx = \int_1^R (f(x))^2 \,\text dx+ \int_R^\infty \underbrace {(f(x))^2}_{\leqslant f(x)} \,\text dx$
  $\leqslant \int_1^R (f(x))^2 \,\text dx+ \int_R^\infty f(x) \,\text dx$
  $<\infty$ .
• FALSE. Let such that for $n \in \mathbb N$ . Let $f\left(n \pm \frac{1}{n^2}\right) =0$ and extend linearly between the intervals $\left[ n- \frac{1}{n^2},n \right ]$ and $\left[ n,n + \frac{1}{n^2}\right ]$ . Put elsewhere. The function looks like triangles of height with base $\frac{2}{n^2}$ centered at . Then
  $\int_1^\infty f(x) \,\text dx = \frac 14 + \sum_{n=2}^\infty \frac {\text{base}\times\text{height}}{2} = \frac 14 + \sum_{n=2}^\infty \frac{1}{n^2}<\infty$ converges
  but
  $\sum_{n=1}^\infty f(n) = \sum_{n=1}^\infty 1 = \infty$ does not.
• FALSE. Let $f(x) = |\sin(\pi x)|$ . Then $\int_1^\infty f(x) \,\text dx = \int_1^\infty |\sin (\pi x)| \,\text dx = \infty$ but $\sum_{n=1}^\infty f(n) = \sum_{n=1}^\infty |\sin (\pi n)| = 0$ .