# Understand the problem

For each natural number k, choose a complex number \(z_k\) , with\( | z_k | = 1 \), and denote \( a_k \) by the area of the triangle formed by \(z_k , i \cdot z_k , z_k + i \cdot z_k \) . Then which of the following is true for the series below: $$ \Sigma (a_k)^k $$

(1) It converges only if every \(z_k\) lies in the same quadrant

(2) It always diverges

(3) It always converges

(4) None of the above

##### Source of the problem

##### Topic

Complex Numbers and Geometry

##### Difficulty Level

6 out of 10

##### Suggested Book

Complex Numbers from A to Z by Titu Andreescu

# Start with hints

# Connected Program at Cheenta

# I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

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