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This question has been asked and answered here:limit when it exists of a complex number raised to an integral power but I don't understand why this is so and I can't comment on the original question.

Can someone explain this to me?

my specific problem is finding $$ \lim \limits_{n\to\infty} c_n $$ with $$ c_n= \left(\frac 14\sqrt 2+\frac 14\sqrt 2i\right)^n $$

I see that $$c_n= \left(\frac 14\sqrt 2+\frac 14\sqrt 2i\right)^n = \left(\frac 12\left(\cos\left(\frac \pi 4\right)+i\sin\left(\frac\pi4\right)\right)\right)^n $$

but not why $$ \lvert c_n\rvert= \left(\frac 12\left\lvert \cos\left(\frac \pi 4\right)+i\sin\left(\frac\pi4\right)\right\rvert\right)^n=\left(\frac12\right)^n$$

I feel like I could probably answer a question on a test but I don't understand why it's like this

Cheers Andy

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Andy Grey
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  • Hint: $\cos^2 x + \sin^2 x = 1$ – gammatester Mar 31 '16 at 09:17
  • When you multiply complex numbers together, you can multiply their magnitudes and add their moduli. So, you're right that the magnitude of $C_n$ is $(\frac{1}{2})^n$. However, I'm not sure what you did with the middle step there. – Kaynex Mar 31 '16 at 09:20
  • The expression $\cos(x)+i\sin(x)$ is a point on the unit circle. Knowing that you can conclude that it has a magnitude of 1. – MrYouMath Mar 31 '16 at 09:21
  • @kaynex that was taken from the solution. I'm studying in German so I'm not sure what the moduli are but I'll look it up – Andy Grey Mar 31 '16 at 09:45

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