A Ratio of Sums | Problem 323

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[MrGeorge]

I thought of the second method, too, but was not sure if it would work because of | i | not being less than one i.e. the sum does not converge.

[vzaimo]

Hi, Sybermath! Thank you for video! Here's an equation from Russian bear: n = 1+2i-3-4i+...+n*i^(n-1). Could you solve, please? 🐻

[over]

Isn't this a special case of one we did a while ago, based on 2n/n, where n=7?

[SidneiMV]

i⁰iⁿ + i¹iⁿ + i²iⁿ + i³iⁿ = iⁿ(1 + i - 1 - i) = 0

(i + i - 1 - i) = C = 0

i⁰C + i⁴C + i⁸C + ... + i²⁴C + i²⁸ = i²⁸
i⁰C + i⁴C + i⁸C + i¹²C - i¹⁵ = -i¹⁵

i²⁸/(-i¹⁵) = -i¹³ = -i¹²i = *-i*