Answered: Σε k=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section: Chapter Questions

Problem 57RE

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Question

Determine whether the following series converge. Justify your answers.

Expert Solution

Step 1: Given,

The series $\sum_{k = 1}^{\infty} e^{- k^{3}} .$ We have to determine whether the series is convergent or divergent.

Step 2: Concept Used

Cauchy Root Test:

Suppose that we have the series $\sum_{}^{} a_{n} .$ Define,

$L = \lim_{n \to \infty} \sqrt[n]{|a_{n}|} = \lim_{n \to \infty} {|a_{n}|}^{\frac{1}{n}}$

if $L < 1$ the series is absolutely convergent (and hence convergent).
if $L > 1$ the series is divergent.
if $L = 1$ the series may be divergent, conditionally convergent, or absolutely convergent.

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