# Show that,#sqrt(-2+2sqrt(-2+2sqrt(-2+2sqrt(-2+.............))))=1+-i#?

##### 1 Answer

Converges to

#### Explanation:

Let

First, Assuming that this infinite series converges (i.e. assuming S exists and takes the value of a complex number),

And if you solve for S:

and applying the quadratic formula you get:

Usually the square root function takes the positive value thus

Thus, if it converges then it must converge to

Now all you have to do is prove that it converges or if you are lazy like me then you can plug

I repeated this many times on my Ti - 83 and found that it does get closer for example after I repeated it somewhere like 20 times I got approximately

pretty good approximation