Please Help With Summations!

$\sum_{k=1}^{\infty} \ln \frac{k+1}{k}$

$\sum_{k=1}^{\infty} \ln (k+1) - \ln{k}$

$\not\ln \not 2 ~- ~\ln 1$
$\not\ln \not 3~ - ~\not\ln \not 2$
$\not\ln \not 4 ~-~ \not\ln \not 3$
$\not\ln \not 5 ~-~ \not\ln \not 4$
............
$\ln \infty ~-~ \not{...}$

It makes,
$\ln \infty - \ln 1 = \infty$

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$\sum_{k=1}^{\infty} \frac{1}{1 + 2+3+4+...+k}$

$\sum_{k=1}^{\infty} \frac{1}{\frac{k(k+1)}{2}}$

$\sum_{k=1}^{\infty} \frac{2}{k(k+1)}$

$\sum_{k=1}^{\infty} \frac{2}{k} - \frac{2}{k+1}$

$\frac{2}{1} - \not\frac{2}{2}$
$\not\frac{2}{2} - \not\frac{2}{3}$
$\not\frac{2}{3} - \not\frac{2}{4}$
$\not\frac{2}{4} - \not\frac{2}{5}$
............
$\not\frac{1}{\infty} - \frac{1}{\infty}$

It makes,
$\frac{2}{1} - \frac{1}{\infty} = 2$