Category:Examples of Use of Binomial Theorem
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This category contains examples of use of the Binomial Theorem.
Let $X$ be one of the standard number systems $\N$, $\Z$, $\Q$, $\R$ or $\C$.
Let $x, y \in X$.
Then:
\(\ds \forall n \in \Z_{\ge 0}: \, \) | \(\ds \paren {x + y}^n\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k\) | |||||||||||
\(\ds \) | \(=\) | \(\ds x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 3} } {3!} x^{n - 3} y^3 + \cdots\) |
where $\dbinom n k$ is $n$ choose $k$.
Pages in category "Examples of Use of Binomial Theorem"
The following 21 pages are in this category, out of 21 total.
B
- Binomial Theorem/Examples
- Binomial Theorem/Examples/(1 + x)^7
- Binomial Theorem/Examples/11^4
- Binomial Theorem/Examples/4th Power of Difference
- Binomial Theorem/Examples/4th Power of Sum
- Binomial Theorem/Examples/5th Power of Difference
- Binomial Theorem/Examples/5th Power of Sum
- Binomial Theorem/Examples/6th Power of Difference
- Binomial Theorem/Examples/6th Power of Sum
- Binomial Theorem/Examples/Cube of Difference
- Binomial Theorem/Examples/Cube of Sum
- Binomial Theorem/Examples/Cube of Sum/Corollary
- Binomial Theorem/Examples/Square Root of 2