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Fungrim entry: 34136c

1x+a1xa2(xa)3/2\left|\frac{1}{\sqrt{x + a}} - \frac{1}{\sqrt{x}}\right| \le \frac{\left|a\right|}{2 {\left(x - \left|a\right|\right)}^{3 / 2}}
Assumptions:x(0,)andaRanda<xx \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left|a\right| < x
TeX:
\left|\frac{1}{\sqrt{x + a}} - \frac{1}{\sqrt{x}}\right| \le \frac{\left|a\right|}{2 {\left(x - \left|a\right|\right)}^{3 / 2}}

x \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left|a\right| < x
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("34136c"),
    Formula(LessEqual(Abs(Sub(Div(1, Sqrt(Add(x, a))), Div(1, Sqrt(x)))), Div(Abs(a), Mul(2, Pow(Sub(x, Abs(a)), Div(3, 2)))))),
    Variables(x, a),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(a, RR), Less(Abs(a), x))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC