Unit 4 · Lesson 6 of 9

A root test exposes an exponential factor.

An nth root can separate exponential growth from slower polynomial effects. Its limiting value decides most cases.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its finite nth-root term value.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase term index n? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

nth rootLimit · dashed

Term index n10 Exponential factor a0.8

finite nth-root term value

\sqrt[10]{|u_10|}=\frac{0.8}{10^{2/10}}=0.504766,\quad L=0.8

Do not use a finite root value as the limit. At limit one, return to another test instead of guessing.

Assumptions, units, and limits

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Take the nth root of absolute term magnitude. The polynomial factor tends to one after this root because its logarithm divided by n tends to zero. At root limit one, the test gives no conclusion; use comparison or another method.

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Keep the reasoning visible

n^{2/n}=e^{2\ln n/n}\to1

Do not use a finite root value as the limit. At limit one, return to another test instead of guessing.

What makes this model useful?

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

Do not replace the limiting test with a finite root sample. In this example $a=1$ converges by the p-series test even though the root test is inconclusive.

A worked example

Calculate the starting scenario

Use term index n = 10 ; exponential factor a = 0.8 .
$\sqrt[10]{|u_10|}=\frac{0.8}{10^{2/10}}=0.504766,\quad L=0.8$
Do not use a finite root value as the limit. At limit one, return to another test instead of guessing. The checked quantity is finite nth-root term value.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the finite nth-root term value using term index n 10 , exponential factor a 0.8 . State the meaning of its sign and units.

Checked quantity: finite nth-root term value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

nth rootLimit · dashed

Term index n10 Exponential factor a0.8

finite nth-root term value

\sqrt[10]{|u_10|}=\frac{0.8}{10^{2/10}}=0.504766,\quad L=0.8

Do not use a finite root value as the limit. At limit one, return to another test instead of guessing.

Assumptions, units, and limits

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Calculate and explain

Apply the idea to a new scenario

For $n=50$ and $a=1,$ find the nth-root value and state what the root test alone concludes.

Checked quantity: finite nth-root term value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

nth rootLimit · dashed

Term index n50 Exponential factor a1

finite nth-root term value

\sqrt[50]{|u_50|}=\frac{1}{50^{2/50}}=0.855148,\quad L=1

Do not use a finite root value as the limit. At limit one, return to another test instead of guessing.

Assumptions, units, and limits

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

u_n=\frac{a^n}{n^2},\quad\sqrt[n]{|u_n|}=\frac{|a|}{n^{2/n}}\to|a|

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

nth rootLimit · dashed

Term index n10 Exponential factor a0.8

finite nth-root term value

\sqrt[10]{|u_10|}=\frac{0.8}{10^{2/10}}=0.504766,\quad L=0.8

Do not use a finite root value as the limit. At limit one, return to another test instead of guessing.

Assumptions, units, and limits

Positive a. Root limit <1 proves absolute convergence; >1 proves divergence; =1 is inconclusive, and this particular $a=1$ series converges by the p-series test.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Take the nth root of absolute term magnitude. The polynomial factor tends to one after this root because its logarithm divided by n tends to zero. At root limit one, the test gives no conclusion; use comparison or another method.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.