Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use subsequential limits to disprove convergence.
- Justify the conclusion "The full sequence does not converge" using the stated assumptions.
Before you start
Sequences and parity.
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
Does aₙ=(−1)ⁿ+1/n converge?
Why this math matters
Use subsequential limits to disprove convergence. 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:
- n runs through positive integers.
- Convergence means a single real limit.
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.
Detect nonconvergence through incompatible subsequences
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Does aₙ=(−1)ⁿ+1/n converge?
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
a₂ₖ=1+1/(2k)→1
Even indices form a subsequence approaching one.
Work through the mathematics
a₂ₖ₊₁=−1+1/(2k+1)→−1
Odd indices approach a different value.
Check the conclusion
The full sequence does not converge
Every subsequence of a convergent sequence must share its limit.
The result
The full sequence does not converge
Every subsequence of a convergent sequence must share its limit.
Common mistakes to catch
- Boundedness is weaker than convergence.
- Checking only one subsequence can miss another limiting behavior.
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
What are limsup and liminf?
Show a hint
Use the persistent upper and lower cluster values.
Reveal answer and explanation
limsup=1, liminf=−1
The small positive correction vanishes along either parity.
Practice 2
Is the sequence bounded?
Show a hint
Bound the alternating part and 1/n.
Reveal answer and explanation
Yes
Boundedness allows convergent subsequences but does not force the entire sequence to converge.
Take the idea with you
Use alternating simulation outputs to distinguish convergence from persistent oscillation.
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: Distinguish cancellation from absolute summability
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