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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Check convergence and quantify how much a finite truncation misses.
- Justify the conclusion "The omitted tail is (1/3)⁵/(1−1/3)=1/162" using the stated assumptions.
Before you start
Finite geometric sums and limits.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find Σₙ₌₀∞(1/3)ⁿ and the error after terms n=0 through n=4.
Why this math matters
Check convergence and quantify how much a finite truncation misses. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The ratio is constant with magnitude below one in the worked series.
- The partial sum includes both displayed endpoint indices.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Control the tail of an infinite geometric sum
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find Σₙ₌₀∞(1/3)ⁿ and the error after terms n=0 through n=4.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
For |r|<1, partial sums equal (1−rᴺ⁺¹)/(1−r)
The finite formula isolates the term that tends to zero.
Work through the mathematics
At r=1/3, the infinite sum is 1/(1−1/3)=3/2
The power tail vanishes in the limit.
Check the conclusion
The omitted tail is (1/3)⁵/(1−1/3)=1/162
The tail itself is another geometric sum beginning at n=5.
The result
The omitted tail is (1/3)⁵/(1−1/3)=1/162
The tail itself is another geometric sum beginning at n=5.
Common mistakes to catch
- The number of terms is one more than the last index when starting at zero.
- A convergent total and a useful approximation error are separate quantities.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does Σ2ⁿ converge?
Show a hint
Its terms do not tend to zero.
Reveal answer and explanation
No
The geometric convergence condition fails.
Practice 2
How many terms beginning at n=0 are needed for tail below 0.01?
Show a hint
For N terms the tail is (1/3)ᴺ/(2/3).
Reveal answer and explanation
Five terms
Four leave 1/54>0.01; five leave 1/162<0.01.
Take the idea with you
Choose a truncation length from an explicit tail bound instead of visual stabilization.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Solve a recurrence with two independent growth modes
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