ArithmeticGeometricMean[1, b] \[Proportional] Pi/(2 (Log[4] - Log[b])) + ((Pi (1 + Log[b/4]))/(8 (Log[4] - Log[b])^2)) b^2 + (b^4 Pi (8 - 40 Log[2]^2 + Log[1024] + 5 (-1 + Log[256] - 2 Log[b]) Log[b]))/(256 (Log[4] - Log[b])^3) + (b^6 Pi (24 + 4 Log[2] (27 + (23 - 132 Log[2]) Log[2]) + Log[b] (-54 - 92 Log[2] + 792 Log[2]^2 + Log[b] (23 - 396 Log[2] + 66 Log[b]))))/(3072 (Log[4] - Log[b])^4) + (b^8 Pi (8 (96 + Log[2] (744 + Log[2] (2230 + (1487 - 11256 Log[2]) Log[2]))) + Log[b] (4 (-744 + Log[2] (-4460 - 4461 Log[2] + 45024 Log[2]^2)) + Log[b] (4460 + 6 (1487 - 22512 Log[2]) Log[2] + (-1487 + 45024 Log[2] - 5628 Log[b]) Log[b]))))/ (393216 (Log[4] - Log[b])^5) + (b^10 Pi (16 (240 + Log[2] (2640 + Log[2] (12765 + Log[2] (31565 + (17153 - 165480 Log[2]) Log[2])))) + Log[b] (8 (-2640 + Log[2] (-25530 + Log[2] (-94695 - 68612 Log[2] + 827400 Log[2]^2))) + Log[b] (12 (4255 + Log[2] (31565 + 34306 Log[2] - 551600 Log[2]^2)) + Log[b] (-63130 + 8 Log[2] (-17153 + 413700 Log[2]) + Log[b] (17153 - 827400 Log[2] + 82740 Log[b]))))))/ (7864320 (Log[4] - Log[b])^6) + O[b^12/Log[b]] /; (b -> 0)

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</mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 551600 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 34306 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 31565 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4255 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( 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2001-10-29