Review: p-Series and Conditional/Absolute Convergence

How confident are you with using the p-Series Test?

• If every question on the exam used the p-Series Test, I would get a 5.

• I feel like I know how to use the test, but get confused when applying it to problems sometimes.

• I don't know how to use the test at all.

The p-Series Test states that a series in the form  $\sum_{n=1}^{\infty}\frac{1}{n^{\ p}}=\frac{1}{1^{\ p}}+\frac{1}{2\ ^p}+\frac{1}{3^{\ p}}+...+\frac{1}{n^{\ p}}+...$ :

• Converges if  $p>1$  

• Converges if  $0<p\le1$  

• Diverges if  $p>1$  

• Diverges if  $0<p\le1$ 

According to the p-Series Test, the series  $\sum_{n=1}^{\infty}\frac{1}{n^{\frac{1}{3}}}$ :

• Converges

• Diverges

• Inconclusive (or test can't be applied)

According to the p-Series Test, the series  $\sum_{n=1}^{\infty}\frac{-2}{n^3}$ :

• Converges

• Diverges

• Inconclusive (or test can't be applied)

According to the p-Series Test, the series  $\sum_{n=1}^{\infty}n^{\frac{4}{3}}$ :

• Converges

• Diverges

• Inconclusive (or test can't be applied)

According to the p-Series Test, the series  $\sum_{n=1}^{\infty}\left(\frac{1}{n^3}+\frac{1}{n^e}\right)$ :

• Converges

• Diverges

• Inconclusive (or test can't be applied)

According to the p-Series Test, the series  $\sum_{n=1}^{\infty}n^{-2}\cdot\sqrt{n}$ :

• Converges

• Diverges

• Inconclusive (or test can't be applied)

Do you remember how to find absolute and conditional convergence?

• Yes

• No

 $\sum_{n=1}^{\infty}a_n$  is absolutely convergent if  $\sum_{n=1}^{\infty}\left|a_n\right|$ converges. 

 $\sum_{n=1}^{\infty}a_n$  is conditionally convergent if  $\sum_{n=1}^{\infty}a_n$ converges but  $\sum_{n=1}^{\infty}\left|a_n\right|$ diverges.

This means that you will need to solve for the convergence of both series (original and absolute value). Does this make sense?

• Yes

• No

Using any of the tests we've learned so far (nth Term Test, AST, and p-Series Test), the series  $\sum_{n=1}^{\infty}\frac{\left(-1\right)^n}{n^{\frac{3}{2}}}$ :

• Converges conditionally

• Converges absolutely

• Diverges

• Inconclusive (or test can't be applied)