X x n x
However, for an arbitrary number r, one can define from both sides of the result, the ordinary binomial theorem is recovered.
For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine.
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Visit Stack Exchange I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$.
According to De Moivre's formula, Indeed, since each term of the binomial expansion is an increasing function of n, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to e.
The binomial theorem is closely related to the probability mass function of the negative binomial distribution.
However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.
is a specific positive integer known as a binomial coefficient.
(When an exponent is zero, the corresponding power expression is taken to be 1 and this multiplicative factor is often omitted from the term.
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Hence one often sees the right side written as The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical.