$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\bigmath }[1]{\displaystyle #1} \newcommand {\choicebreak }[0]{} \newcommand {\type }[0]{} \newcommand {\notes }[0]{} \newcommand {\keywords }[0]{} \newcommand {\offline }[0]{} \newcommand {\comments }[0]{\begin {feedback}} \newcommand {\multiplechoice }[0]{\begin {multipleChoice}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Exercises relating to the Ratio and Root Tests.

Note: As always, you can type the word “infinity” or “infty” (without quotes) in any entry box to indicate that the answer is infinite.
Consider the infinite series Apply the Ratio Test: Apply the Root Test: Both tests indicate that the series convergesdiverges .
Oftentimes the Ratio Test is easier to apply than the Root Test when dealing with factorials. Use the Ratio Test to determine convergence or divergence of the series In the spaces below, record the eponymous “ratio” in the first blanks, simplify it in the second blanks, and then record the limit. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Oftentimes the Root Test is easier to apply when terms have large exponents (growing faster than a constant times $n$). Use the Root Test to determine the convergence or divergence of the series First say what should quantity should have its $n$-th root taken, then simplify, and lastly record the value of the limit. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .
Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive .

Sample Quiz Questions

Determine which of the following three infinite series will lead to inconclusive results for the Ratio Test and then determine whether that series is convergent or divergent.

I inconclusive, converges I inconclusive, diverges II inconclusive, converges II inconclusive, diverges III inconclusive, converges III inconclusive, diverges