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A226985 Sum of inverse of increasing integers with a difference of 0, 1, 2, 3, ...: 1 + 1/2 + 1/4 + 1/7 + 1/11 + 1/16 + 1/22 + 1/29 + 1/37 + .... 10

%I

%S 2,3,7,3,6,5,4,6,7,5,4,4,0,1,0,7,7,6,4,3,2,1,6,8,6,1,2,2,2,3,7,4,3,2,

%T 4,5,1,9,1,3,8,0,5,9,0,9,4,0,6,7,1,2,0,2,9,6,7,3,3,1,3,3,8,9,1,2,5,1,

%U 1,3,6,4,7,1,0,4,5,9,2,1,3,8,9,4,1,6,3,9,7,6,6,8,2,7,8,2,9,6,7,7,5,3,3,3,3,9

%N Sum of inverse of increasing integers with a difference of 0, 1, 2, 3, ...: 1 + 1/2 + 1/4 + 1/7 + 1/11 + 1/16 + 1/22 + 1/29 + 1/37 + ....

%C This is a convergent series since the denominator is quadratic.

%C We can note that tanh(sqrt(7)*Pi/2) = 0.9995... which is close to 1 by 0.05% so this constant is very close to 2*Pi/sqrt(7). - _Didier Guillet_, Jul 12 2013

%F Sum_{k >= 1} 1/(1+k*(k-1)/2).

%F It equals 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7). - _Giovanni Resta_, Jun 26 2013

%e 2.3736546754401077643216861222374324519138059094067120296733133891251...

%t RealDigits[2*Pi*Tanh[Sqrt[7]*Pi/2]/Sqrt[7], 10, 110][[1]] (* _Giovanni Resta_, Jun 26 2013 *)

%o (PARI) sumpos(k=1,1/(1+k*(k-1)/2)) \\ _Charles R Greathouse IV_, Jun 26 2013

%o (PARI) 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) \\ _Charles R Greathouse IV_, Jun 26 2013

%Y Cf. A000124.

%K nonn,cons

%O 1,1

%A _Didier Guillet_, Jun 25 2013

%E a(12)-a(87) from _Giovanni Resta_, Jun 26 2013

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Last modified March 26 10:18 EDT 2019. Contains 321491 sequences. (Running on oeis4.)