Kenneth ChengMathematical modeling
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Infinite geometric series and error control

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

Revisit first: Geometric sequences

TOPIC 01

Infinite geometric series and error control

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

What you will be able to explain

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

Model or definition

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

S∞=a/(1−q)(∣q∣<1)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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does a negative ratio prevent an infinite geometric sum?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

First six geometric terms: 2, 1, 0.5, 0.25, 0.125, 0.0625; sum=3.9375. The vertical display scale adjusts to include all six terms. Infinite sum=4 because |r|<1.

First six geometric terms: 2, 1, 0.5, 0.25, 0.125, 0.0625; sum=3.9375. The vertical display scale adjusts to include all six terms. Infinite sum=4 because |r|<1.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare q=−1/2 and q=−1 by their partial sums and remainders.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the infinite geometric sum.

a1=2,q=1/2a_1=2,\quad q=1/2
  • 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
Use this intermediate relation.
S∞=a/(1−q)S∞=a/(1−q)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    S=2/(1−1/2)=4S=2/(1-1/2)=4

  3. The ratio magnitude is less than one.

The requested value is 4.

Checks and common pitfalls: The ratio magnitude is less than one.

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

02 / Standard#Worked example

Find the tail after the first 3 terms of a geometric series.

a1=3,q=1/2a_1=3,\quad q=1/2
  • 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
Tail=a q³/(1−q)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    R3=3(1/8)/(1/2)=3/4R_3=3(1/8)/(1/2)=3/4

  3. The tail begins at the fourth term, aq³.

The requested value is 0.75.

Checks and common pitfalls: The tail begins at the fourth term, aq³.

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

03 / Transfer#Worked example

Evaluate the convergent telescoping series.

∑n=1∞4n(n+1)\sum_{n=1}^{\infty}\frac{4}{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

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

  2. Apply the stated relation and retain its conditions.

    SN=4(1−1/(N+1))S_N=4(1-1/(N+1))

  3. Apply the stated relation and retain its conditions.

    N→∞⇒S=4N\to\infty\Rightarrow S=4

  4. First justify a finite telescoping identity, then take its limit.

The requested value is 4.

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

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

04 / Foundation#Your turn

Find the infinite geometric sum.

a1=5,q=1/2a_1=5,\quad q=1/2
  • 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
Use this intermediate relation.
S∞=a/(1−q)S∞=a/(1−q)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    S=5/(1−1/2)=10S=5/(1-1/2)=10

  3. The ratio magnitude is less than one.

The requested value is 10.

Checks and common pitfalls: The ratio magnitude is less than one.

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

05 / Foundation#Your turn

Find the infinite sum with alternating signs.

a1=6,q=−1/2a_1=6,\quad q=-1/2
  • 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
Use this intermediate relation.
S∞=a/(1−q)S∞=a/(1−q)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    S=6/(1+1/2)=12/3S=6/(1+1/2)=12/3

  3. A negative ratio is permitted when its magnitude is below one.

The requested value is 4.

Checks and common pitfalls: A negative ratio is permitted when its magnitude is below one.

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

06 / Foundation#Your turn

Does this geometric series converge? Explain.

a1=7,q=−1a_1=7,\quad q=-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
Inspect even and odd partial sums.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    S2m=0,S2m+1=7S_{2m}=0,\quad S_{2m+1}=7

  3. Two different subsequence limits show nonconvergence.

The requested relation or conclusion is shown below.

divergent\text{divergent}

Checks and common pitfalls: Two different subsequence limits show nonconvergence.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

07 / Foundation#Your turn

Express the repeating decimal 0.dddd… as a fraction value, where the single digit d repeats.

d=9d=9
  • 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
Use this intermediate relation.
x=d/10+d/100+⋯x=d/10+d/100+⋯
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    x=(9/10)/(1−1/10)=9/9x=(9/10)/(1-1/10)=9/9

  3. A one-digit block repeats at ratio 1/10.

The requested value is 1.

Checks and common pitfalls: A one-digit block repeats at ratio 1/10.

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

08 / Standard#Your turn

Express the repeating decimal 0.dddd… as a fraction value, where the single digit d repeats.

d=1d=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
Use this intermediate relation.
x=d/10+d/100+⋯x=d/10+d/100+⋯
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    x=(1/10)/(1−1/10)=1/9x=(1/10)/(1-1/10)=1/9

  3. A one-digit block repeats at ratio 1/10.

The requested value is 0.111111111111.

Checks and common pitfalls: A one-digit block repeats at ratio 1/10.

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

09 / Standard#Your turn

Find the tail after the first 3 terms of a geometric series.

a1=10,q=1/2a_1=10,\quad q=1/2
  • 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
Tail=a q³/(1−q)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    R3=10(1/8)/(1/2)=10/4R_3=10(1/8)/(1/2)=10/4

  3. The tail begins at the fourth term, aq³.

The requested value is 2.5.

Checks and common pitfalls: The tail begins at the fourth term, aq³.

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

10 / Standard#Your turn

Find the least n so the tail of 1+1/2+1/4+… is at most 2^(1−t).

t=11t=11
  • 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
Use this intermediate relation.
Rn=21−nR_n=2^{1−n}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    21−n≤21−11⇒n≥112^{1-n}\le2^{1-11}\Rightarrow n\ge11

  3. The tail is positive, so it equals the absolute truncation error.

The requested value is 11.

Checks and common pitfalls: The tail is positive, so it equals the absolute truncation error.

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

11 / Standard#Your turn

Evaluate the convergent telescoping series.

∑n=1∞12n(n+1)\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

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

  2. Apply the stated relation and retain its conditions.

    SN=12(1−1/(N+1))S_N=12(1-1/(N+1))

  3. Apply the stated relation and retain its conditions.

    N→∞⇒S=12N\to\infty\Rightarrow S=12

  4. 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.

12 / Transfer#Your turn

Find the least n so the tail of 1+1/2+1/4+… is at most 2^(1−t).

t=13t=13
  • 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
Use this intermediate relation.
Rn=21−nR_n=2^{1−n}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    21−n≤21−13⇒n≥132^{1-n}\le2^{1-13}\Rightarrow n\ge13

  3. The tail is positive, so it equals the absolute truncation error.

The requested value is 13.

Checks and common pitfalls: The tail is positive, so it equals the absolute truncation error.

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

13 / Transfer#Your turn

Evaluate the convergent telescoping series.

∑n=1∞14n(n+1)\sum_{n=1}^{\infty}\frac{14}{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

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

  2. Apply the stated relation and retain its conditions.

    SN=14(1−1/(N+1))S_N=14(1-1/(N+1))

  3. Apply the stated relation and retain its conditions.

    N→∞⇒S=14N\to\infty\Rightarrow S=14

  4. First justify a finite telescoping identity, then take its limit.

The requested value is 14.

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

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • 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?

    Board plan

    • Model or definition: Check convergence before summing and interpret recurring decimals and truncation errors.
      S∞=a/(1−q)(∣q∣<1)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.

    Anticipated thinking

    • An oscillating partial-sum sequence with q=−1 does not converge.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗