Question

Find the value of
$lo{g}_{4}2-lo{g}_{8}2+lo{g}_{16}\mathrm{2...}\infty$

Answer 1

$\Rightarrow lo{g}_{4}2-lo{g}_{8}2+lo{g}_{16}\mathrm{2...}\infty$
$\Rightarrow lo{g}_{(2{)}^2}2-lo{g}_{(2{)}^3}2+lo{g}_{(2{)}^4}\mathrm{2...}\infty$
$\Rightarrow \frac{1}{2}-\frac{1}{3}+\frac{1}{4}...\infty$
$\Rightarrow (\frac{1}{2}+\frac{1}{4}...\infty )-(\frac{1}{3}+\frac{1}{3}...\infty )$.