A323737 Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j)) - k as k approaches infinity.

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

Offset

0,1

Comments

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

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

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

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

...

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

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.

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.

Links

Table of n, a(n) for n=0..86.

Formula

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

Example

0.68538530275123375551874460543013077034359371013804...

Crossrefs

Cf. A323736.

Sequence in context: A209283 A209285 A199171 * A374883 A350860 A010501

Adjacent sequences: A323734 A323735 A323736 * A323738 A323739 A323740

Keyword

nonn,cons

Author

Jon E. Schoenfield, Feb 06 2019

Status

approved