Question

√x+√x+√x+...=√x√x√x .... then what is the value of x

Answer 1

$\sqrt x+\sqrt x+\sqrt x+\ldots=\sqrt x \sqrt x \sqrt x\ldots$

The obvious solution is $x=0$.

For $x\in(0,1)$ the series on the left side diverges and the product on the right side converges and its limit is 0. Therefore there is no solution on that interval.

For $x=1$, again, the series on the left side diverges, but the right side is equal to 1. Therefore

For $x\in(1,\infty)$ both, the series on the left side and the product on the right side of the equation, approach infinity, but it doesn't really mean they're equal for any $x\in(1,\infty)$.

If we consider their limits though, then they're indeed equal.