W07 Stepwise

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

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:

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

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02

05

Limits and convergence

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

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

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

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03

04

General term of a sequence

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

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

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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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