Power Series — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{2}^{-n}{\left(x\right)}^n$$

   Answer: $2$

   Step-by-step solution

   1. $$L=\frac{|x|}{2}<1$$
   2. $$R=2$$
2. 2.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{3}^{-n}{\left(x-1\right)}^n$$

   Answer: $3$

   Step-by-step solution

   1. $$L=\frac{|x-1|}{3}<1$$
   2. $$R=3$$
3. 3.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{4}^{-n}{\left(x+2\right)}^n$$

   Answer: $4$

   Step-by-step solution

   1. $$L=\frac{|x+2|}{4}<1$$
   2. $$R=4$$
4. 4.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{5}^{-n}{\left(x-3\right)}^n$$

   Answer: $5$

   Step-by-step solution

   1. $$L=\frac{|x-3|}{5}<1$$
   2. $$R=5$$
5. 5.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{2}^n{\left(x\right)}^n$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$L=\frac{|x|}{\frac{1}{2}}<1$$
   2. $$R=\frac{1}{2}$$
6. 6.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{6}^{-n}{\left(x+1\right)}^n$$

   Answer: $6$

   Step-by-step solution

   1. $$L=\frac{|x+1|}{6}<1$$
   2. $$R=6$$
7. 7.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{7}^{-n}{\left(x\right)}^n$$

   Answer: $7$

   Step-by-step solution

   1. $$L=\frac{|x|}{7}<1$$
   2. $$R=7$$
8. 8.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{3}^{-n}{\left(x-2\right)}^n$$

   Answer: $3$

   Step-by-step solution

   1. $$L=\frac{|x-2|}{3}<1$$
   2. $$R=3$$
9. 9.

   Find the radius of convergence.

   $$\sum _{n=0}^{\infty }{5}^{-n}{\left(x+3\right)}^n$$

   Answer: $5$

   Step-by-step solution

   1. $$L=\frac{|x+3|}{5}<1$$
   2. $$R=5$$
10. 10.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }{8}^{-n}{\left(x-1\right)}^n$$

    Answer: $8$

    Step-by-step solution

    1. $$L=\frac{|x-1|}{8}<1$$
    2. $$R=8$$
11. 11.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }{2}^{-n}{\left(x-4\right)}^n$$

    Answer: $2$

    Step-by-step solution

    1. $$L=\frac{|x-4|}{2}<1$$
    2. $$R=2$$
12. 12.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }{9}^{-n}{\left(x\right)}^n$$

    Answer: $9$

    Step-by-step solution

    1. $$L=\frac{|x|}{9}<1$$
    2. $$R=9$$
13. 13.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }10^{-n}{\left(x+2\right)}^n$$

    Answer: $10$

    Step-by-step solution

    1. $$L=\frac{|x+2|}{10}<1$$
    2. $$R=10$$
14. 14.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }{4}^{-n}{\left(x-5\right)}^n$$

    Answer: $4$

    Step-by-step solution

    1. $$L=\frac{|x-5|}{4}<1$$
    2. $$R=4$$
15. 15.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }{3}^n{\left(x\right)}^n$$

    Answer: $\frac{1}{3}$

    Step-by-step solution

    1. $$L=\frac{|x|}{\frac{1}{3}}<1$$
    2. $$R=\frac{1}{3}$$
16. 16.

    Find the radius of convergence.

    $$\sum _{n=0}^{\infty }12^{-n}{\left(x-2\right)}^n$$

    Answer: $12$

    Step-by-step solution

    1. $$L=\frac{|x-2|}{12}<1$$
    2. $$R=12$$

More from Calculus II

• Integration by Parts
• Trigonometric Integrals
• Partial Fractions
• Improper Integrals
• Sequence Limits
• Geometric Series
• p-Series
• Convergence Tests
• Taylor Series
• Parametric Derivatives
• Parametric Arc Length
• Polar Area