Question

Decide if this sequence is convergent or divergent:
√ n√ n√ n√ [ Cos n + i sin n ]
, n≥ 2


Note :
Please more than method
the above means
,third times powered to ^1/n
third root of n

Answer 1

Answer on Question #50129, Math, Complex Analysis

Given:

$$\sum_{n=2}^{\infty} \sqrt[2]{\sqrt[n]{\sqrt[n]{\cos n} + i \sin n}}$$

Decide:

if these series is convergent or divergent

Solution:

$$a_n = (\cos n + i \sin n)^{\frac{1}{2n^3}} = (e^{in})^{\frac{1}{2n^3}} = e^{i \cdot \frac{\arg(e^{in}) + 2\pi k}{2n^3}} \quad k \in \mathbb{Z}$$ $$\arg(e^{in}) = n$$ $$|a_n| = 1$$

we can use the fact

$$\lim_{n \to \infty} a_n = 0 \quad \Leftrightarrow \quad \lim_{n \to \infty} |a_n| = 0$$

we obtain

$\lim_{n \to \infty} |a_n| = 1 \neq 0$ the necessary condition of convergence of series is not executed, so the series is divergent.

Answer: divergent

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