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Sequences and their limits
What it means for aₙ to approach L, and the limits that define e and π.
A sequence converges to L when its terms eventually stay as close to L as you like. That “as you like” is the ε in every analysis proof. (1 + 1/n)ⁿ creeps up to e; n/(n + 1) creeps up to 1; (−1)ⁿ never settles. The graph shows the terms approaching the limit.
Ví dụ đã làm · limit of (1 + 1/n)^n as n -> oo
Limit of (1 + 1/n)^n as n → oo
Bước một bước
- \lim_{n \to \infty^+-} \left(1 + \frac{1}{n}\right)^{n}
Try direct substitution first.
- \
As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- = e
Take the limit.
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Làm thế nào: Sequences and their limits
- Try direct substitution first.
- As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- Take the limit.
Câu hỏi người ta hỏi
What is the ε, δ definition actually saying?
That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.
Why does the harmonic series diverge when its terms go to zero?
Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.
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