Answered: find the limit of the sequence and find…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.2: Sequences, Series And Summation Notation

Problem 19E: Find the next term of each sequence 1, 3, 6, 10,.

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find the limit of the sequence and find if the sequence converge.

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Expert Solution

Step 1

Consider the limit

$\{\frac{6}{n e^{- n}}\}$

Step 2

$\begin{array}{l} \lim_{n \to \infty} \frac{6}{n e^{- n}} = \lim_{n \to \infty} \frac{6 e^{n}}{n} (∵ \frac{1}{a^{- b}} = a^{b}) \\ = \lim_{n \to \infty} \frac{6 e^{n}}{n} (\frac{\infty}{\infty}) \\ Apply L'hospital's rule \\ = \lim_{n \to \infty} \frac{(6 e^{n})'}{(n)'} \\ = \lim_{n \to \infty} \frac{6 e^{n}}{1} \\ = \lim_{n \to \infty} 6 e^{n} \\ = \lim_{n \to \infty} 6 e^{\infty} \\ = \infty \\ \lim_{n \to \infty} \frac{6}{n e^{- n}} = \infty \end{array}$

$\lim_{n \to \infty} \frac{6}{n e^{- n}} = \infty$

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