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[tex] \sqrt{2 + \sqrt{2 + \sqrt{2........... \infty } } } [/tex]
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Answer this
[tex] \sqrt{2 + \sqrt{2 + \sqrt{2........... \infty } } } [/tex]
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So,
x = √2+√2+√2+...+
Now let's square both sides and let's see what we get,
x^2 = 2+√2+√2+√2+...+
Now, we know that this series is being continued till infinity that means we can again call all the terms coming under root to be x as a whole.
x^2 = 2*(√2+√2+√2+√2+...+)
x^2 = 2x
x^2 - 2x = 0
x(x-2) = 0
So, therefore x=0 or x = 2
However, it obviously cannot be zero because that whole thing under root will surely give us some real value and not zero.
So, we neglect x = 0 and thus, x = 2 that means
√2+√2+√2+√2+...+ = 2