LEARN · EXPLAIN · REVISE

Evaluate the convergent telescoping series.

Read the idea, work independently, then explain what changed.

T04高二理組數學思維本(2026).pdf · III. Series; infinite geometric series · PDF 35 / printed page 34

TOPIC 01

Infinite geometric series and error control

Check convergence before summing and interpret recurring decimals and truncation errors.

Check convergence before summing and interpret recurring decimals and truncation errors.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

\sum_{n=1}^{\infty}\frac{12}{n(n+1)}

For nonzero a, convergence requires |q|<1; a finite partial sum formula does not by itself establish an infinite sum.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Finite partial sum=t(1−1/(N+1)).

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$S_N=12(1-1/(N+1))$
Apply the stated relation and retain its conditions.
$N\to\infty\Rightarrow S=12$
First justify a finite telescoping identity, then take its limit.

The requested value is 12.

Checks and common pitfalls: First justify a finite telescoping identity, then take its limit.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Check convergence before summing and interpret recurring decimals and truncation errors.
Which condition is essential in infinite geometric series and error control?
Does a negative ratio prevent an infinite geometric sum?

Model or definition: Check convergence before summing and interpret recurring decimals and truncation errors.
$S_\infty=a/(1-q)\quad(|q|<1)$
Conditions: For nonzero a, convergence requires |q|<1; a finite partial sum formula does not by itself establish an infinite sum.