From Integrable Functions To Limits Of Sequences

Table 1:

f : 0,1 R	Sequence n2·xn	Sequence yn	$\underset{n \to \infty}{l i m}$ xn	$\underset{n \to \infty}{l i m}$ yn
1	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{1}{n})}$	$\prod_{k = 1}^{n} (1 + \frac{1}{n})$	1	e
x	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{1}{k})}$	$\prod_{k = 1}^{n} (1 + \frac{k}{n^{2}})$	$\frac{1}{2}$	$\sqrt{e}$
xp, pR $_{+}^{*}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{n^{p - 1}}{k^{p}})}$	$\prod_{k = 1}^{n} (1 + \frac{k^{p}}{n^{p + 1}})$	$\frac{1}{p + 1}$	e $\frac{1}{p + 1}$
ln(1+x)	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{1}{n}}{\ln (1 + \frac{k}{n})})}$	$\prod_{k = 1}^{n} [1 + \frac{1}{n} \cdot \ln (1 + \frac{k}{n})]$	2·ln2-1	$\frac{4}{e}$
x·ln(1+x)	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{1}{k}}{\ln (1 + \frac{k}{n})})}$	$\prod_{k = 1}^{n} [1 + \frac{k}{n^{2}} \cdot \ln (1 + \frac{k}{n})]$	$\frac{1}{4}$	e $\sqrt[4]{e}$
$\frac{\ln (1 + x)}{1 + x}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{n + k}{n^{2}}}{\ln (1 + \frac{k}{n})})}$	$\prod_{k = 1}^{n} [1 + \frac{1}{n + k} \cdot \ln (1 + \frac{k}{n})]$	$\frac{\ln^{2} 2}{2}$	$\sqrt{2^{\ln 2}}$
$\frac{1}{\sqrt{1 + x^{2}}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\sqrt{n^{2} + k^{2}}}{n^{2}})}$	$\prod_{k = 1}^{n} [1 + \frac{1}{\sqrt{n^{2} + k^{2}}}]$	ln( $\sqrt{2}$ +1)	$\sqrt{2}$ +1
$\frac{x}{\sqrt{1 + x^{2}}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\sqrt{n^{2} + k^{2}}}{n \cdot k})}$	$\prod_{k = 1}^{n} [1 + \frac{k}{\sqrt{n^{2} + k^{2}}}]$	$\sqrt{2}$ -1	e $\sqrt{2} - 1$
arctgx	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{1}{n}}{a r c t g \frac{k}{n}})}$	$\prod_{k = 1}^{n} [1 + \frac{1}{n} \cdot a r c t g \frac{k}{n}]$	$\frac{π}{4}$ - $\frac{\ln 2}{2}$	$\frac{\sqrt[4]{4 \cdot e^{π}}}{2}$
x·arctgx	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{1}{k}}{a r c t g \frac{k}{n}})}$	$\prod_{k = 1}^{n} [1 + \frac{k}{n^{2}} \cdot a r c t g \frac{k}{n}]$	$\frac{π}{4}$ - $\frac{1}{2}$	$\sqrt[4]{e^{π - 2}}$
x·ex	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{\frac{1}{k}}{e^{\frac{k}{n}}})}$	$\prod_{k = 1}^{n} [1 + \frac{k}{n^{2}} \cdot e^{\frac{k}{n}}]$	1	e
$\sqrt{1 + x^{2}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{1}{\sqrt{n^{2} + k^{2}}})}$	$\prod_{k = 1}^{n} [1 + \frac{\sqrt{n^{2} + k^{2}}}{n^{2}}]$	$\frac{\sqrt{2} + \ln (\sqrt{2} + 1)}{2}$	$\sqrt{e^{\sqrt{2}} \cdot (\sqrt{2} + 1)}$
x· $\sqrt{1 + x^{2}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{n}{k \cdot \sqrt{n^{2} + k^{2}}})}$	$\prod_{k = 1}^{n} [1 + \frac{k \cdot \sqrt{n^{2} + k^{2}}}{n^{3}}]$	$\frac{2 \cdot \sqrt{2} - 1}{3}$	$\sqrt[3]{e^{\sqrt{2} - 1}}$
$\sqrt{1 - x^{2}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{1}{\sqrt{n^{2} - k^{2}}})}$	$\prod_{k = 1}^{n} [1 + \frac{\sqrt{n^{2} - k^{2}}}{n^{2}}]$	$\frac{π}{4}$	$\sqrt[4]{e^{π}}$
x· $\sqrt{1 - x^{2}}$	$\sum_{k = 1}^{n} \frac{1}{\ln (1 + \frac{n}{k \cdot \sqrt{n^{2} - k^{2}}})}$	$\prod_{k = 1}^{n} [1 + \frac{k \cdot \sqrt{n^{2} - k^{2}}}{n^{3}}]$	$\frac{1}{3}$	$\sqrt[3]{e}$

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