Answered: Find the limit of the following…

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 34E: Find the sum of the first five terms of the sequence with the given general term. an=2n

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Question

Find the limit of the following sequences or determine that the sequence diverges.

{ ((n)/(n + 5))ⁿ}

Expert Solution

Step 1

Given:

The given series is $a_{n} = {(\frac{n}{(n + 5)})}^{n}$

The series is divergent if $\lim_{n \to \infty} a_{n} \neq 0$ else the series is convergent.

Step 2

Determine the limit of the series.

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} ({(\frac{n}{(n + 5)})}^{n})$

Since the series is of indeterminate forms. Therefore Apply L-hopital's rule.

$\begin{array}{rcl} \lim_{n \to \infty} (\frac{n}{{(n + 5)}^{n}}) & = & \lim_{n \to \infty} e^{\ln ({(\frac{n}{n + 5})}^{n})} \\ = & \lim_{n \to \infty} e^{n \ln (\frac{n}{n + 5})} \\ = & e^{\lim_{n \to \infty} n \ln (\frac{n}{n + 5})} \\ = & e^{\lim_{n \to \infty} \frac{\ln (\frac{n}{n + 5})}{\frac{1}{n}}} \end{array}$

Since it is an indeterminant form. Apply L hopital's rule.

$\begin{array}{rcl} e^{\lim_{n \to \infty} \frac{\ln (\frac{n}{n + 5})}{\frac{1}{n}}} & = & e^{\lim_{n \to \infty} \frac{\frac{d}{d n} (\ln (\frac{n}{n + 5}))}{\frac{d}{d n} (\frac{1}{n})}} \\ = & e^{\lim_{n \to \infty} (- n (n + 5) (- \frac{n}{{(n + 5)}^{2}} + \frac{1}{n + 5}))} \\ = & e^{\lim_{n \to \infty} (\frac{- 5 n}{n + 5})} \\ = & e^{(- 5 \lim_{n \to \infty} \frac{n}{n + 5})} \end{array}$

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