Does the Sum Converge?

Calculus Level 3

Let θ k \theta_k be a randomly generated real number which varies with k k . Does the following sum converge?

∑ k = 0 ∞ ( 4 3 ) k e ( − k 3 + i θ k ) \Large \sum^\infty_{k=0}\left(\frac{4}{3}\right)^{k} \, e^{(-\frac{k}{3} + i \theta_k)}

Notation : i = − 1 i=\sqrt{-1} denotes the imaginary unit .

Yes No

Steven Chase by Steven Chase , 37, USA

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1 solution

Tom Engelsman
May 2, 2018

The above infinite series can be written as :

∑ k = 0 ∞ ( 4 3 ) k e ( − k 3 + i θ k ) = ∑ k = 0 ∞ ( 4 3 ⋅ e 1 / 3 ) k ( c o s ( θ k ) + i ⋅ s i n ( θ k ) ) \Large \sum^\infty_{k=0}\left(\frac{4}{3}\right)^{k} \, e^{(-\frac{k}{3} + i \theta_k)} = \Large \sum^\infty_{k=0} (\frac{4}{3 \cdot e^{1/3}})^{k} (cos(\theta_{k}) + i \cdot sin(\theta_{k}))

Given that ∣ 4 3 ⋅ e 1 / 3 ∣ = 0.9553 < 1 |\frac{4}{3 \cdot e^{1/3}}| = 0.9553 < 1 and that − 1 ≤ c o s ( θ k ) , s i n ( θ k ) ≤ 1 -1 \le cos(\theta_{k}), sin(\theta_{k}) \le 1 for all k k , the series converges.

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