Sequences and Series — Quiz 2

Questions in this quiz (10)

1. Determine if the series $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt[6]{4n+1}}$$ converges.

   • Converges absolutely
   • Converges conditionally
   • Diverges
   • Converges by ratio test

2. Find the radius of convergence for the power series $$\sum_{n=0}^{\infty}\frac{(-1)^n x^n}{3n^2+2n+1}$$.

   • $$R=1$$
   • $$R=3$$
   • $$R=\frac13$$
   • $$R=\infty$$

3. Find the radius of convergence for $$\sum_{n=0}^{\infty}\frac{x^{3n}}{n!}$$.

   • $$R=1$$
   • $$R=3$$
   • $$R=\infty$$
   • $$R=\frac13$$

4. Find the radius of convergence for $$\sum_{n=1}^{\infty}\frac{n!(x-2)^n}{3^n}$$.

   • $$R=0$$
   • $$R=1$$
   • $$R=3$$
   • $$R=\infty$$

5. The series $$\sum_{n=0}^{\infty}\frac{nx^n}{5^n}$$ converges on what interval?

   • $$(-5,5)$$
   • $$[-5,5]$$
   • $$(-5,5]$$
   • $$[-5,5)$$

6. If $$f(x)=(x-2)+\frac{(x-2)^2}{2}+\frac{(x-2)^3}{3}+\cdots$$, give the interval of convergence for $$f''(x)$$.

   • $$(-3,1)$$
   • $$(-1,3)$$
   • $$(1,5)$$
   • $$(1,3)$$

7. Determine a power series for $$\cos(x^2)$$.

   • $$\sum_{n=0}^{\infty}\frac{(-1)^n x^{4n}}{(2n)!}$$
   • $$\sum_{n=0}^{\infty}\frac{(-1)^n x^{4n}}{(2n)!}$$
   • $$\sum_{n=0}^{\infty}\frac{(-1)^n x^{4n+2}}{(2n+1)!}$$
   • $$\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{n!}$$

8. Let $$P(x)=3+2(x-1)-4(x-1)^2+6(x-1)^3$$ be a Taylor polynomial of degree $$3$$ for $$f(x)$$ centered about $$1$$. What is $$f''(1)$$?

   • $$-8$$
   • $$-4$$
   • $$-8$$
   • $$8$$

9. Create a Maclaurin series for $$g(x)=\sin(e^x)$$.

   • $$\sin(1)+\cos(1)x+\frac{\cos(1)-\sin(1)}{2}x^2+\cdots$$
   • $$\sin(1)+\cos(1)x+\frac{\cos(1)}{2}x^2+\cdots$$
   • $$\sin(1)+e^x\cos(1)+\cdots$$
   • $$\sin(x)+\cos(1)x+\cdots$$

10. Estimate the value of $$\sqrt{1.2}$$ using the third-degree Taylor polynomial for $$y=\sqrt{x}$$ centered about $$x=1$$.

    • $$1.095$$
    • $$1.100$$
    • $$1.105$$
    • $$1.095$$