This Limit Looks Impossible… Until e Appears! Exponential Growth Explained
Автор: Chan Lye Lee
Загружено: 2026-02-22
Просмотров: 79
Описание:
In this video, we evaluate the limit
[
\lim_{n\to\infty} \frac{1^n + 2^n + 3^n + \cdots + n^n}{n^n}
]
At first glance, it looks messy. But with a clever transformation and the classic exponential limit
[
\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n = e^x,
]
the expression simplifies beautifully.
This problem connects:
• exponential growth
• dominance of largest terms
• geometric series ideas
• the constant (e)
• asymptotic thinking
Perfect for students preparing for:
SMO, AMC, AIME, AP Calculus, IB HL, Olympiad training, and advanced high school mathematics.
If you enjoy limits involving exponential expressions and elegant transformations, this one is for you.
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