MathAdmin: Created page with " Which of the following sequences $(a_n)_{n\ge 1}$ converges? Which diverges? Give reasons for your answ..."

2017-03-19T18:38:27Z

Created page with " Which of the following sequences <math style="vertical-align: -5px">(a_n)_{n\ge 1}</math> converges? Which diverges? Give reasons for your answ..."

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Which of the following sequences  <math style="vertical-align: -5px">(a_n)_{n\ge 1}</math>  converges? Which diverges? Give reasons for your answers!

(a)  <math style="vertical-align: -15px">a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>

(b)  <math style="vertical-align: -15px">a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''L'Hôpital's Rule'''
|-
|
        Suppose that  <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>  and  <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>  are both zero or both  <math style="vertical-align: -1px">\pm \infty .</math>
|-
|
        If  <math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math>  is finite or  <math style="vertical-align: -4px">\pm \infty ,</math>
|-
|
        then  <math style="vertical-align: -19px">\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}\,=\,\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|Let
        <math>\begin{array}{rcl}
\displaystyle{y} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg(1+\frac{1}{2n}\bigg)^n.}
\end{array}</math>
|-
|We then take the natural log of both sides to get
|-
|        <math>\ln y = \ln\bigg(\lim_{n\rightarrow \infty} \bigg(1+\frac{1}{2n}\bigg)^n\bigg).</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|We can interchange limits and continuous functions.
|-
|Therefore, we have
|-
|
        <math>\begin{array}{rcl}
\displaystyle{\ln y} & = & \displaystyle{\lim_{n\rightarrow \infty} \ln \bigg(1+\frac{1}{2n}\bigg)^n}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} n\ln\bigg(1+\frac{1}{2n}\bigg)}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{\ln \bigg(1+\frac{1}{2n}\bigg)}{\frac{1}{n}}.}
\end{array}</math>
|-
|Now, this limit has the form  <math style="vertical-align: -13px">\frac{0}{0}.</math>
|-
|Hence, we can use L'Hopital's Rule to calculate this limit.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
|Now, we have
|-
|
        <math>\begin{array}{rcl}
\displaystyle{\ln y } & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{\ln \bigg(1+\frac{1}{2n}\bigg)}{\frac{1}{n}}}\\
&&\\
& = & \displaystyle{\lim_{x\rightarrow \infty} \frac{\ln \bigg(1+\frac{1}{2x}\bigg)}{\frac{1}{x}}}\\
&&\\
& \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\frac{2x}{2x+1}\big(\frac{-1}{2x^2}\big)}{\big(-\frac{1}{x^2}\big)}}\\
&&\\
& = & \displaystyle{\lim_{x\rightarrow \infty} \frac{1}{2}\bigg(\frac{2x}{2x+1}\bigg)}\\
&&\\
& = & \displaystyle{\frac{1}{2}.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
|-
|Since  <math style="vertical-align: -5px">\ln y= 1/2,</math>  we know
|-
|        <math>y=e^{1/2}.</math>
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First, we have
|-
|       <math>\lim_{n\rightarrow \infty}\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n=\lim_{n\rightarrow \infty} (-1)^n\bigg(\frac{1+n}{n}\bigg)^n.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Now, let
|-
|        <math>\begin{array}{rcl}
\displaystyle{y} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg(\frac{1+n}{n}\bigg)^n.}
\end{array}</math>
|-
|We then take the natural log of both sides to get
|-
|        <math>\ln y = \ln\bigg(\lim_{n\rightarrow \infty} \bigg(\frac{1+n}{n}\bigg)^n\bigg).</math>
|-
|We can interchange limits and continuous functions.
|-
|Therefore, we have
|-
|
        <math>\begin{array}{rcl}
\displaystyle{\ln y} & = & \displaystyle{\lim_{n\rightarrow \infty} \ln \bigg(\frac{1+n}{n}\bigg)^n}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} n\ln\bigg(\frac{1+n}{n}\bigg)}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{\ln \bigg(\frac{1+n}{n}\bigg)}{\frac{1}{n}}.}
\end{array}</math>
|-
|Now, this limit has the form  <math style="vertical-align: -13px">\frac{0}{0}.</math>
|-
|Hence, we can use L'Hopital's Rule to calculate this limit.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
|Now, we have
|-
|
        <math>\begin{array}{rcl}
\displaystyle{\ln y } & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{\ln \bigg(\frac{1+n}{n}\bigg)}{\frac{1}{n}}}\\
&&\\
& = & \displaystyle{\lim_{x\rightarrow \infty} \frac{\ln \bigg(\frac{1+x}{x}\bigg)}{\frac{1}{x}}}\\
&&\\
& \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\frac{x}{1+x}\big(\frac{-1}{x^2}\big)}{\big(-\frac{1}{x^2}\big)}}\\
&&\\
& = & \displaystyle{\lim_{x\rightarrow \infty} \frac{x}{1+x}}\\
&&\\
& = & \displaystyle{1.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
|-
|Since  <math style="vertical-align: -4px">\ln y= 1,</math>  we know
|-
|        <math>y=e.</math>
|-
|Since
|-
|       <math>\lim_{n\rightarrow \infty} \bigg(\frac{1+n}{n}\bigg)^n\neq 0,</math>
|-
|we have
|-
|
        <math>\begin{array}{rcl}
\displaystyle{\lim_{n\rightarrow \infty} a_n} & = & \displaystyle{\lim_{n\rightarrow \infty} (-1)^n\bigg(\frac{1+n}{n}\bigg)^n}\\
&&\\
& = & \displaystyle{\text{DNE}.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|    '''(a)'''    <math>e^{1/2}</math>
|-
|    '''(b)'''    <math>\text{DNE}</math>
|}
[[009C_Sample_Final_3|'''Return to Sample Exam''']]

009C Sample Final 3, Problem 1 - Revision history

MathAdmin: Created page with " Which of the following sequences (a_n)_{n\ge 1} converges? Which diverges? Give reasons for your answ..."

MathAdmin: Created page with " Which of the following sequences $(a_n)_{n\ge 1}$ converges? Which diverges? Give reasons for your answ..."