Functions of Number Theory - 27.7 Lambert Series as Generating Functions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
27.7.E4	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)\frac{x^{n}}{1-x^{n}}=\frac{x}{(1-x)^{2}}}}$ \sum_{n=1}^{\infty}\Eulertotientphi[]@{n}\frac{x^{n}}{1-x^{n}} = \frac{x}{(1-x)^{2}}	${\displaystyle{\displaystyle|x|<1}}$	sum(phi(n)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = (x)/((1 - x)^(2))	Sum[EulerPhi[n]*Divide[(x)^(n),1 - (x)^(n)], {n, 1, Infinity}, GenerateConditions->None] == Divide[x,(1 - x)^(2)]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
27.7.E5	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)x^{n}}}$ \sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}} = \sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}x^{n}	${\displaystyle{\displaystyle|x|<1}}$	sum((n)^(alpha)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = sum(add(divisors(alpha))*(x)^(n), n = 1..infinity)	Error	Failure	Missing Macro Error	Failed [3 / 3] Result: 2.671514971 Test Values: {alpha = 3/2, x = 1/2} Result: 1.507450946 Test Values: {alpha = 1/2, x = 1/2} ... skip entries to safe data	-