General Binomial Theorem/Convergence
Theorem
Recall the General Binomial Theorem:
| \(\ds \paren {1 + x}^\alpha\) | \(=\) | \(\ds 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots\) |
The above binomial series:
- converges when $\size x < 1$
- diverges when $\size x > 1$
For the special case where $x = 1$, the binomial series converges if $n > -1$.
For the special case where $x = -1$, the binomial series converges if $n > 0$.
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial theorem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): binomial series (expansion)
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): binomial series (expansion)