Complex number 2!

Calculus Level 4

For an integer p p , define f p ( α ) = e i α / p 2 . e 2 i α / p 2 . e 3 i α / p 2 . . . . . . . . e p i α / p 2 f_p(\alpha) = e^{i\alpha /p^2} . e^{2i\alpha /p^2} . e^{3i\alpha /p^2} ........ e^{pi\alpha /p^2}

Evaluate lim ⁡ n → ∞ f n ( π ) \displaystyle \lim_{n \to \infty} f_n(\pi) .

i i 1 1 − i -i − 1 -1 None of These

Nishant Rai by Nishant Rai , 25, India

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1 solution

Chew-Seong Cheong
May 30, 2015

f p ( α ) = ∏ k = 1 p e i k α p 2 = e i α p 2 ∑ k = 1 p k = e i α p ( p + 1 ) 2 p 2 = e i α ( 1 2 + 1 2 p ) f_p(\alpha) = \prod_{k=1}^p {e^{i\frac{k\alpha}{p^2}}} = e^{i \frac{\alpha}{p^2} \sum_{k=1}^p {k}} = e^{i\frac{\alpha p(p+1)}{2p^2}} = e^ {i\alpha \left(\frac{1}{2}+\frac{1}{2p}\right)}

⇒ lim ⁡ n → ∞ f n ( π ) = e i π ( 1 2 + 0 ) = cos ⁡ π 2 + i sin ⁡ π 2 = 0 + i \Rightarrow \lim_{n \to \infty} {f_n(\pi)} = e^ {i \pi \left( \frac{1}{2}+0\right)} = \cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}} = 0 + \boxed{i}

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Moderator note:

Simple standard appraoch

@Tanishq Varshney

here is the solution.

Nishant Rai - 6 years ago

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Later i was able to solve it, i missed the two.

Tanishq Varshney - 6 years ago

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