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| | <span class="exam">Test if the following sequences converge or diverge. Also find the limit of each convergent sequence. | | <span class="exam">Test if the following sequences converge or diverge. Also find the limit of each convergent sequence. |
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| − | ::<span class="exam">a) <math style="vertical-align: -12px">a_n=\frac{\ln(n)}{\ln(n+1)}</math>
| + | <span class="exam">(a) <math style="vertical-align: -16px">a_n=\frac{\ln(n)}{\ln(n+1)}</math> |
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| − | ::<span class="exam">b) <math style="vertical-align: -12px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>
| + | <span class="exam">(b) <math style="vertical-align: -15px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009C Sample Final 2, Problem 1 Solution|'''<u>Solution</u>''']] |
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| − | |'''L'Hopital's Rule''' | |
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| − | Suppose that <math>\lim_{x\rightarrow \infty} f(x)</math> and <math>\lim_{x\rightarrow \infty} g(x)</math> are both zero or both <math style="vertical-align: -1px">\pm \infty .</math>
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| − | If <math>\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math> is finite or <math style="vertical-align: -4px">\pm \infty ,</math>
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| − | then <math>\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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| − | '''(a)''' | + | [[009C Sample Final 2, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | !Step 1:
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| − | !Step 2:
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| − | '''(b)'''
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| − | !Step 1:
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| − | !Step 2:
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| − | !Final Answer:
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| − | | '''(a)'''
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| − | | '''(b)'''
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| | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.
(a) 
(b) 
Solution
Detailed Solution
Return to Sample Exam