W07 Stepwise

Due date: Friday 2/26, 11:59pm

Sequences

01

01

L’Hopital practice - converting indeterminate form

By imitating the technique of the L’Hopital’s Rule example, find the limit of the sequence:

$$a_n=\sqrt{n}\ln (1+\frac{1}{n})$$

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Solution

Solutions---0140-01

02

05

Limits and convergence

For each sequence, either write the limit value (if it converges), or write ‘diverges’.

(a) ${\displaystyle \frac{5n-1}{12n+9}}$ $$ (b) ${\displaystyle {(-1)}^n\left(\frac{5n-1}{12n+9}\right)}$ $$ (c) ${\displaystyle \sqrt{4+\frac{1}{n}}}$

(d) ${\cos }^{-1}\left({\displaystyle \frac{n^3}{n^3+1}}\right)$ $$ (e) ${\displaystyle 10+{(-\frac{1}{9})}^n}$ $$

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Solution

Solutions---0140-05

03

04

General term of a sequence

Find a formula for the general term (the $n^{\text{th}}$ term) of each sequence:

(a) ${\displaystyle \frac{1}{1},\frac{-1}{8},\frac{1}{27},\dots }$ $$ (b) ${\displaystyle \frac{2}{6},\frac{3}{7},\frac{4}{8},\dots }$ $$ (c) ${\displaystyle \frac{3}{5},-\frac{4}{25},\frac{5}{125},-\frac{6}{625},\dots }$

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Solution

Solutions---0140-04

Series

04

01

General term of a series

Write this series in summation notation:

$$\frac{1}{1}-\frac{2^2}{2\cdot 1}+\frac{3^3}{3\cdot 2\cdot 1}-\frac{4^4}{4\cdot 3\cdot 2\cdot 1}+\cdots$$

(Hint: Find a formula for the general term $a_n$.)

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Solution

Solutions---0150-01