Infinite Nested Imaginary
Find The Expression
Today’s Challenge
My first course at Warwick doesn’t start until October. In the meantime, let’s try and find the expression of the above-nested radicals with the imaginary number.
Give the problem a try before jumping in for the solution!
To start off, we let
As the highlighted part goes to infinity, we can thus replace it with the complex number z
Upon squaring and rearranging, we can turn it into a complex quadratic
Using the good ol’ quadratic formula, we get
We have ignored the minus sign because we only consider the principal square root.
Now, before you think we have found our answer. Recall that if a complex number is within a square root, we can find that expression in the root-free form!
To do this, we will turn the complex number into the polar form and apply De Moivre’s theorem
where
and
and
Therefore
Applying De Moivre’s theorem and double angle formulas for sine and cosine, we finally get
Alternatively, one can set the square root of (1 + 4i) = a + bi and solve for the coefficients a and b
Finally, substituting the above and simplifying, we arrive at
Well done!
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Happy reading,
Barry 🍩