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