Let P(x)=1+e2 ix+2e3 ix. Evaluate NN-1 k=0N-1P(k2).

GATE 2023 MA – Question 13

Real Analysis · Sequences and Series of Functions · 1 mark · Multiple choice

Let $P(x)=1+e^{2\pi ix}+2e^{3\pi ix}$. Evaluate $\lim_{N\to\infty}N^{-1}\sum_{k=0}^{N-1}P(k\sqrt2)$.

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Correct answer: (B) 1

Explanation

For either phase a=e^(2πi√2) or e^(3πi√2), a≠1 and |a|=1. Its geometric sum divided by N is (1−a^N)/(N(1−a))→0. The constant term has average 1. **B**.

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