one term
🔢CALCULUS II STUDY GUIDE
Sequences and Limits
Find sequence limits using algebra, monotonicity, bounds, squeeze reasoning, and recurrence relations.
8 key ideas8 flashcards5 practice questions
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finite sequence limit
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
A sequence assigns a number aₙ to each positive integer n.
- 02
Sequence convergence means aₙ approaches one finite number as n grows.
- 03
Rational expressions in n can often be divided by the highest power of n.
- 04
Exponential growth outruns polynomial growth for bases greater than one.
- 05
A bounded monotone sequence converges.
- 06
An increasing bounded sequence approaches its least upper bound.
- 07
The squeeze theorem applies when lower and upper sequences meet at one limit.
- 08
A recursive sequence can be tested for monotonicity, boundedness, and a limit that satisfies its recurrence.
PART OF THE CALCULUS II GUIDE
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Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→Bounded means convergent
USE THIS INSTEAD
A bounded sequence can oscillate forever.
NOT QUITE
→A recursive fixed point proves convergence
USE THIS INSTEAD
It supplies candidates only; convergence still needs proof.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What does aₙ → L mean?Show answer ＋
Terms become arbitrarily close to L for all sufficiently large n.
2What limit does 1/n have?Show answer ＋
Zero.
3What limit does rⁿ have when |r|<1?Show answer ＋
Zero.
4Does (−1)ⁿ converge?Show answer ＋
No, because it alternates between two numbers.
5What guarantees convergence for a monotone sequence?Show answer ＋
A bound in the direction of motion.
6How are rational sequence limits simplified?Show answer ＋
Divide numerator and denominator by the highest power of n present.
7Can every bounded sequence converge?Show answer ＋
No. It may oscillate.
8How can a recursive limit be tested?Show answer ＋
Assume convergence, take limits on both sides, solve for candidates, then verify.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Find lim n/(n+1).
Divide by n to obtain 1/(1+1/n), which tends to 1.
02Why does n²/2ⁿ tend to zero?
Exponential growth with base 2 eventually exceeds polynomial growth.
03Why is (−1)ⁿ/n convergent?
Its absolute magnitude is 1/n, which tends to zero, so squeezing gives zero.
04What else is needed after solving a recursive fixed-point equation?
Proof that the sequence converges and reaches the selected fixed point.
05How do boundedness and monotonicity work together?
Monotonicity prevents oscillation, while the bound prevents escape to infinity.
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