Convergence of sequences and series
Lesson 5 of 10
The epsilon definition of a limit
Objective. State and use the formal definition of sequence convergence.
The key idea
aₙ → L means: for every ε > 0 there is N such that |aₙ − L| < ε for all n > N. Smaller ε simply forces a larger N.
Worked example
Show 1/n → 0 using the definition.
- 1|1/n − 0| = 1/n.
- 2We need 1/n < ε, i.e. n > 1/ε.
- 3So take N = ⌈1/ε⌉.
Answer: For any ε > 0, N = ⌈1/ε⌉ works
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Choosing N before ε, reversing the order of the quantifiers.
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