Problems

01

L’Hopital practice - converting indeterminate form

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

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

02

Squeeze theorem

Determine whether the sequence converges, and if it does, find its limit:

(a) $a_n={\displaystyle \frac{{\cos }^{2}n}{2^n}}$ $$ (b) $b_n={\displaystyle {(2^n+3^n)}^{1/n}}$

(Hint for (b): Verify that $3\le b_n\le {(2\cdot 3^n)}^{1/n}$.)

03

Computing the terms of a sequence

Calculate the first four terms of each sequence from the given general term, starting at $n=1$:

(a) $\cos \pi n$ $$ (b) ${\displaystyle \frac{n!}{2^n}}$ $$ (c) ${(-1)}^{n+1}$ $$ (d) ${\displaystyle \frac{n}{n+1}}$ $$ (e) ${\displaystyle \frac{3^n}{n!}}$ $$ (f) ${\displaystyle \frac{(2n-1)!}{n!}}$

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) ${\displaystyle \frac{3}{5},-\frac{4}{25},\frac{5}{125},-\frac{6}{625},\dots }$

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

06

Limits and convergence

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

(a) ${1.01}^n$ $$ (b) ${\displaystyle 2^{1/n}}$ $$ (c) ${\displaystyle \frac{n!}{9^n}}$ $$ (d) ${\displaystyle \frac{3n^2+n+2}{2n^2-3}}$

(e) ${\displaystyle \frac{\cos n}{n}}$ $$ (f) $\ln 5^n-\ln n!$ $$ (g) ${\displaystyle {(2+\frac{4}{n^2})}^{1/3}}$ $$ (h) ${\displaystyle n\sin \frac{\pi }{n}}$

07

Limits and convergence

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

(a) ${\displaystyle \ln \left(\frac{2n+1}{3n+4}\right)}$ $$ (b) ${\displaystyle \frac{e^n}{2^n}}$ $$ (c) ${\displaystyle \frac{{(\ln n)}^2}{n}}$ $$ (d) ${\displaystyle \frac{{(-1)}^n{(\ln n)}^2}{n}}$

(e) ${\displaystyle \frac{3-4^n}{2+7\cdot 4^n}}$ $$ (f) ${\displaystyle {(1+\frac{1}{n})}^n}$ $$ (g) ${\displaystyle \frac{1}{\ln (1+\frac{1}{n})}}$