Fungrim entry: b01280

\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

Assumptions:

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

TeX:

\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Factorial`	$n !$	Factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("b01280"),
    Formula(Equal(Div(Pow(Log(Add(1, x)), k), Factorial(k)), Sum(Mul(Mul(Pow(-1, Sub(n, k)), StirlingCycle(n, k)), Div(Pow(x, n), Factorial(n))), Tuple(n, k, Infinity)))),
    Variables(x, k),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(x, CC), Less(Abs(x), 1))))

Topics using this entry

Stirling numbers