Imaginary e e

Algebra Level 3

Let S L = ∑ m = 1 L e ⋅ i m − 1 \displaystyle S_L = \sum_{m=1}^L e \cdot i^{m-1} , where i = − 1 i= \sqrt{-1} and e e denotes Euler's number .

If m m and n n are positive integers larger than 10, find the closed form of S 2 m + S 2 n + 1 S_{2m} + S_{2n+1} to 4 decimal places.


The answer is 2.7182818.

Viki Zeta by Viki Zeta , 19, India

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1 solution

Viki Zeta
Mar 18, 2017

S n = e + e i − e − e i + e + e i − e − e i + … S n = 0 If n is even S n = e If n is odd Hence, S 2 m + S 2 n + 1 = 0 + e = e ≈ 2.7182818 S_n = e + ei - e - ei + e + ei - e - ei + \ldots \\ S_n = 0 ~~ \boxed{\text{If n is even}} \\ S_n = e ~~ \boxed{\text{If n is odd}} \\ \text{Hence, } \\ \boxed{S_{2m} + S_{2n+1} = 0 + e = e \approx 2.7182818}

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