Question

how to find the value?

Answer 1

Using the formula,
$\frac{1}{ log_{ \alpha }( \beta ) } \: = log_{ \beta }( \alpha )$
So,
$\frac{1}{ log_{2}(n) } = log_{n}(2)$
So the series transform into,
$log_{n}(2) + log_{n}(3) + ..... + log_{n}(2017)$
Again using the other formula,
$log( \alpha ) + log( \beta ) = log( \alpha \beta )$
The series again changes into,
$log_{n}(2 \times 3 \times 4....2017) = log_{n}(n)$

Given that 2×3×4....×2017 = 2017! = n
So the required answer for the summation is 1
Hope it may help you !!!!!