009C Sample Midterm 1, Problem 3

Determine whether the following series converges absolutely, conditionally or whether it diverges.

Be sure to justify your answers!

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}}

Foundations:
1. A series $\sum a_{n}$ is absolutely convergent if
the series $\sum |a_{n}|$ converges.
2. A series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum a_n} is conditionally convergent if
the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum |a_n|} diverges and
the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum a_n} converges.

Solution:

Step 1:

Step 2:

Final Answer:

Return to Sample Exam