p-Series — Practice Problems and Worked Solutions

The formulas you need here

17 worked problems

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

1. 1.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^2}$$

   Answer: $\text{Converges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=2\Rightarrow \text{Converges}$$
2. 2.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^3}$$

   Answer: $\text{Converges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=3\Rightarrow \text{Converges}$$
3. 3.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{1}{2}}}$$

   Answer: $\text{Diverges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=\frac{1}{2}\Rightarrow \text{Diverges}$$
4. 4.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^1}$$

   Answer: $\text{Diverges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=1\Rightarrow \text{Diverges}$$
5. 5.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^4}$$

   Answer: $\text{Converges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=4\Rightarrow \text{Converges}$$
6. 6.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{2}{3}}}$$

   Answer: $\text{Diverges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=\frac{2}{3}\Rightarrow \text{Diverges}$$
7. 7.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^5}$$

   Answer: $\text{Converges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=5\Rightarrow \text{Converges}$$
8. 8.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{3}{2}}}$$

   Answer: $\text{Converges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=\frac{3}{2}\Rightarrow \text{Converges}$$
9. 9.

   Determine whether the p-series converges.

   $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{1}{3}}}$$

   Answer: $\text{Diverges}$

   Step-by-step solution

   1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
   2. $$p=\frac{1}{3}\Rightarrow \text{Diverges}$$
10. 10.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^6}$$

    Answer: $\text{Converges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=6\Rightarrow \text{Converges}$$
11. 11.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{4}{5}}}$$

    Answer: $\text{Diverges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=\frac{4}{5}\Rightarrow \text{Diverges}$$
12. 12.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{5}{4}}}$$

    Answer: $\text{Converges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=\frac{5}{4}\Rightarrow \text{Converges}$$
13. 13.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^7}$$

    Answer: $\text{Converges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=7\Rightarrow \text{Converges}$$
14. 14.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{1}{4}}}$$

    Answer: $\text{Diverges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=\frac{1}{4}\Rightarrow \text{Diverges}$$
15. 15.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{9}{8}}}$$

    Answer: $\text{Converges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=\frac{9}{8}\Rightarrow \text{Converges}$$
16. 16.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{10}}$$

    Answer: $\text{Converges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=10\Rightarrow \text{Converges}$$
17. 17.

    Determine whether the p-series converges.

    $$\sum _{n=1}^{\infty }\frac{1}{n^{\frac{7}{8}}}$$

    Answer: $\text{Diverges}$

    Step-by-step solution

    1. $$\sum \frac{1}{n^p}\text{ converges exactly when }p>1.$$
    2. $$p=\frac{7}{8}\Rightarrow \text{Diverges}$$

More from Calculus II

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