Alternating series test Quiz 1

Question 1

Check which of the following series is not convergent.

Accepted Answer

$$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}{{\left[ \sin \left( \dfrac{\pi }{2} \right) \right]}^{2}}}$$

Answer

$$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}\dfrac{1}{n}}$$

Answer

$$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}\dfrac{{{\left( 2n \right)}^{2}}}{{{n}^{2}}+1}}$$

Answer

$$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}\cos \left( \dfrac{\pi }{n} \right)}$$

Question 2

In which of the following series is not absolutely convergent.

Accepted Answer

$$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}n=1+{{2}^{2}}-{{3}^{2}}+{{4}^{2}}+.........}$$

Answer

$$\sum\limits_{n=1}^{\infty }{{{\dfrac{\left( -1 \right)}{n}}^{n}}=-1+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{4}-\dfrac{1}{5}+.............}$$

Answer

$$\sum\limits_{n=1}^{\infty }{{{\dfrac{\left( -1 \right)}{{{n}^{2}}}}^{n}}=1-\dfrac{1}{{{2}^{2}}}+\dfrac{1}{{{3}^{2}}}-\dfrac{1}{{{4}^{2}}}+.......}$$

Answer

None of these

Question 3

What is the value of alternative series at $$n=17$$ and the given series is $$\sum\limits_{n=1}^{\infty }{\dfrac{n\left( n+1 \right)}{{{n}^{2}}}}$$.

Accepted Answer

$$1.059$$

Answer

$$1.05$$

Answer

$$1.065$$

Answer

$$1.95$$

Question 4

Which among the following is true about the given series $$\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}{{\left( n \right)}^{n+\dfrac{1}{2}}}}$$.

Accepted Answer

Diverges

Answer

Converges

Answer

Absolute converges

Answer

Option $$(a)$$ and $$(b)$$ both

Question 5

What is convergence radius if series is $$\sum\limits_{n=1}^{\infty }{\dfrac{6n\left( x+2 \right)\left( n-1 \right)}{2n+1}}$$.

Accepted Answer

$$1$$

Answer

$$2$$

Answer

$$-1$$

Answer

$$6$$