Limit comparison test Quiz 1

Question 1

Check the nature of the series: $$\dfrac{1}{3}+\dfrac{3}{5}+\dfrac{7}{4}+\dfrac{15}{17}+.......$$

Accepted Answer

Divergent

Answer

Convergent

Answer

Conditionally convergent

Answer

Absolutely convergent

Question 2

Determine the nature of $$ \sum\limits_{f=1}^{\infty }{\dfrac{\sqrt{1}}{{{2}^{f}}+f}}$$

Accepted Answer

Convergent

Answer

Cannot be stated

Answer

Divergent

Answer

Both divergent and convergent

Question 3

The following series is: $$\sum\limits_{b=1}^{\infty}{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$

Accepted Answer

Converges

Answer

Diverges

Answer

Conditionally divergent

Answer

Absolutely divergent

Question 4

Use comparison test to determine the nature of $$\sum\limits_{b=1}^{\infty}{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$   and $$ \sum\limits_{w=1}^{\infty }{\dfrac{\sqrt{{{\left( n+1 \right)}^{2}}}}{{{n}^{\dfrac{3}{2}}}}}$$ .

Accepted Answer

$$ \sum\limits_{b=1}^{\infty }{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$ $$	ext{(Convergent) and}$$   $$ \sum\limits_{w=1}^{\infty }{\dfrac{\sqrt{{{\left( n+1 ight)}^{2}}}}{{{n}^{\dfrac{3}{2}}}}}$$ $$	ext{(Divergent)}$$

Answer

$$ \sum\limits_{b=1}^{\infty }{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$ $$	ext{(Convergent) and}$$ $$ \sum\limits_{w=1}^{\infty }{\dfrac{\sqrt{{{\left( n+1 ight)}^{2}}}}{{{n}^{\dfrac{3}{2}}}}}$$ $$	ext{(Convergent)}$$

Answer

$$ \sum\limits_{b=1}^{\infty }{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$ $$	ext{(Divergent) and}$$   $$ \sum\limits_{w=1}^{\infty }{\dfrac{\sqrt{{{\left( n+1 ight)}^{2}}}}{{{n}^{\dfrac{3}{2}}}}}$$ $$	ext{(Convergent)}$$

Answer

$$ \sum\limits_{b=1}^{\infty }{\dfrac{\sqrt{2{{b}^{2}}+4b+1}}{{{b}^{3}}+9}}$$ $$	ext{(Divergent) and}$$   $$ \sum\limits_{w=1}^{\infty }{\dfrac{\sqrt{{{\left( n+1 ight)}^{2}}}}{{{n}^{\dfrac{3}{2}}}}}$$ $$	ext{(Divergent)}$$

Question 5

Determine whether $$\dfrac{1}{{{2}^{3}}}-\dfrac{1}{{{3}^{3}}}(3)+\dfrac{1}{{{4}^{3}}}(6)-\dfrac{1}{{{5}^{3}}}(10)+.....\infty $$  is

Accepted Answer

Conditionally convergent

Answer

Absolutely convergent

Answer

Divergent

Answer

None