Difference between revisions of "009C Sample Midterm 1, Problem 1"

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::<math>a_n=\frac{\ln n}{n}</math>
 
::<math>a_n=\frac{\ln n}{n}</math>
  
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[[009C Sample Midterm 1, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
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!Foundations: &nbsp;
 
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|-
 
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|'''L'Hôpital's Rule, Part 2'''
 
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|
 
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&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g</math>&nbsp; be differentiable functions on the open interval &nbsp;<math style="vertical-align: -5px">(a,\infty)</math>&nbsp; for some value &nbsp;<math style="vertical-align: -4px">a,</math>&nbsp;
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; where &nbsp;<math style="vertical-align: -5px">g'(x)\ne 0</math>&nbsp; on &nbsp;<math style="vertical-align: -5px">(a,\infty)</math>&nbsp; and &nbsp;<math style="vertical-align: -18px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}</math>&nbsp; returns either &nbsp;<math style="vertical-align: -15px">\frac{0}{0}</math>&nbsp; or &nbsp;<math style="vertical-align: -15px">\frac{\infty}{\infty}.</math>&nbsp;
 
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|&nbsp; &nbsp; &nbsp; &nbsp;Then, &nbsp; <math style="vertical-align: -18px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
 
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'''Solution:'''
 
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!Step 1: &nbsp;
 
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|First, notice that
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty} \ln n =\infty</math>
 
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|and
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty} n=\infty.</math>
 
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|Therefore, the limit has the form &nbsp;<math style="vertical-align: -11px">\frac{\infty}{\infty},</math>
 
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|which means that we can use L'Hopital's Rule to calculate this limit.
 
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!Step 2: &nbsp;
 
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|-
 
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|First, switch to the variable &nbsp;<math style="vertical-align: 0px">x</math> &nbsp; so that we have functions and
 
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|can take derivatives. Thus, using L'Hopital's Rule, we have
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{\lim_{n\rightarrow \infty} \frac{\ln n}{n}} & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{\ln x}{x}}\\
 
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&&\\
 
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& \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\big(\frac{1}{x}\big)}{1}}\\
 
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&&\\
 
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& = & \displaystyle{0.}
 
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\end{array}</math>
 
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!Final Answer: &nbsp;
 
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|&nbsp; &nbsp; &nbsp; &nbsp; The sequence converges. The limit of the sequence is &nbsp;<math style="vertical-align: 0px">0.</math>
 
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[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 17:58, 4 November 2017

Does the following sequence converge or diverge?

If the sequence converges, also find the limit of the sequence.

Be sure to jusify your answers!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n=\frac{\ln n}{n}}

(insert picture of handwritten solution)

Detailed Solution for this Problem

Return to Sample Exam

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