If $f$ is positive decreasing and continuous on $[0, \infty)$ I want to show that $\sum_{n=m+1}^{\infty} f(n) \leq \int_m^{\infty} f(t) \, dt$.
123695
Infinite series: Integral test( Proof )
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
16555
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
26541
Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?
81223
Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
20089
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
95575
If $f(s) = (1+s)^{(1+s)^{(1+s)/s}/s}$, show that $\lim_{s \to \infty} f(s)/s = 1$.
4090
Show that $f$ is integrable iff $\sum\limits_{n\in\mathbb{Z}}2^n \mu(\{x\in S:2^n< \left|f(x)\right|\leq2^{n+1}\})<\infty.$
59837
Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r<q$
46101
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
134007
Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.
40790
Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
30940
Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$