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Fungrim entry: 2b2066

∑n=0∞(2nn)xn=11−4x\sum_{n=0}^{\infty} {2 n \choose n} {x}^{n} = \frac{1}{\sqrt{1 - 4 x}}
Assumptions:x∈C  and⁡  ∣x∣<14x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \frac{1}{4}
TeX:
\sum_{n=0}^{\infty} {2 n \choose n} {x}^{n} = \frac{1}{\sqrt{1 - 4 x}}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \frac{1}{4}
Definitions:
Fungrim symbol Notation Short description
Sum∑nf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Infinity∞\infty Positive infinity
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
Abs∣z∣\left|z\right| Absolute value
Source code for this entry:
Entry(ID("2b2066"),
    Formula(Equal(Sum(Mul(Binomial(Mul(2, n), n), Pow(x, n)), For(n, 0, Infinity)), Div(1, Sqrt(Sub(1, Mul(4, x)))))),
    Variables(x),
    Assumptions(And(Element(x, CC), Less(Abs(x), Div(1, 4)))))

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2021-03-15 19:12:00.328586 UTC