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A323737 Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j)) - k as k approaches infinity. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
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 * A010501 A296426 A249282

Adjacent sequences:  A323734 A323735 A323736 * A323738 A323739 A323740

KEYWORD

nonn,cons

AUTHOR

Jon E. Schoenfield, Feb 06 2019

STATUS

approved

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)