Unit 4 · Lesson 4 of 9

A known convergent sum can control a smaller one.

A complicated positive term may be bounded by a simpler power-law term. Use that inequality to decide convergence.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its finite partial sum.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

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

A model you can changeMove one control at a time.

Read the model

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

Target termsConvergent comparison 1/n² · dashed

Terms N10

finite partial sum

S_10=1-\frac1{10+1}=0.909091

Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself.

Assumptions, units, and limits

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

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

From the situation to an equation

Direct comparison transfers convergence from a larger convergent positive series to a smaller one, or divergence from a smaller divergent series to a larger one. Limit comparison works when the ratio approaches a finite positive constant.

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Keep the reasoning visible

\lim_{n\to\infty}\frac{a_n}{b_n}=L\in(0,\infty)

Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself.

What makes this model useful?

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

Inequality direction matters. Smaller than a divergent comparison or larger than a convergent one does not settle the result.

A worked example

Calculate the starting scenario

Use terms n = 10 .
$S_10=1-\frac1{10+1}=0.909091$
Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself. The checked quantity is finite partial sum.

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 partial sum using terms n 10 . State the meaning of its sign and units.

Checked quantity: finite partial sum · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

Target termsConvergent comparison 1/n² · dashed

Terms N10

finite partial sum

S_10=1-\frac1{10+1}=0.909091

Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself.

Assumptions, units, and limits

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

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.

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Calculate and explain

Apply the idea to a new scenario

Find the first 30 terms' sum of 1/[n( $n+1$ )] and compare with its limit.

Checked quantity: finite partial sum · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

Target termsConvergent comparison 1/n² · dashed

Terms N30

finite partial sum

S_30=1-\frac1{30+1}=0.967742

Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself.

Assumptions, units, and limits

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

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.

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

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

0<\frac1{n^2+n}<\frac1{n^2},\quad\sum_{n=1}^N\frac1{n(n+1)}=1-\frac1{N+1}

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

Target termsConvergent comparison 1/n² · dashed

Terms N10

finite partial sum

S_10=1-\frac1{10+1}=0.909091

Being smaller than a convergent positive series proves convergence. Being smaller than a divergent series proves nothing by itself.

Assumptions, units, and limits

Positive terms permit direct comparison. This series also telescopes because 1/[n( $n+1$ )]= $1/n-1$ /( $n+1$ ).

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

The reasoning. Direct comparison transfers convergence from a larger convergent positive series to a smaller one, or divergence from a smaller divergent series to a larger one. Limit comparison works when the ratio approaches a finite positive constant.

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.