login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A323737 Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j)) - k as k approaches infinity. 2

%I

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

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

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

%N Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j)) - k as k approaches infinity.

%C Draw a set of circular arcs, with each arc having its center at the origin and drawn so that it connects the line x=1 to the positive y-axis, and with the j-th arc having radius j; i.e., draw

%C - the 1st arc from (1, 0) to (0, 1),

%C - the 2nd arc from (1, sqrt(3)) to (0, 2),

%C - the 3rd arc from (1, sqrt(8)) to (0, 3),

%C ...

%C - the k-th arc from (1, sqrt(k^2 - 1)) to (0, k).

%C Let alpha(k) be the angle covered by the k-th arc: the first arc is a quarter circle, so alpha(1) = Pi/2; alpha(2) = arcsin(1/2) = 0.523598...; alpha(3) = arcsin(1/3) = 0.339836...; and, in general, alpha(k) = arcsin(1/k). The length of the k-th arc is k*alpha(k) = k*arcsin(1/k), which approaches 1 as k increases.

%C The total length of the first k arcs is then t(k) = Sum_{j=1..k} j*alpha(j) = Sum_{j=1..k} j*arcsin(1/j), which equals k + C - (1/6)/k + (1/12)/k^2 - (19/360)/k^3 + (3/80)/k^4 - (143/5040)/k^5 + (5/224)/k^6 - (731/40320)/k^7 + (35/2304)/k^8 - (4949/380160)/k^9 + (63/5632)/k^10 - ... where C = 0.685385... is the constant whose decimal expansion consists of the terms of this sequence.

%F Lim_{k->infinity} (Sum_{j=1..k} j*arcsin(1/j)) - k.

%e 0.68538530275123375551874460543013077034359371013804...

%Y Cf. A323736.

%K nonn,cons

%O 0,1

%A _Jon E. Schoenfield_, Feb 06 2019

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 3 15:06 EDT 2020. Contains 333197 sequences. (Running on oeis4.)