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A323737
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Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j)) - k as k approaches infinity.
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2
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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Lim_{k->infinity} (Sum_{j=1..k} j*arcsin(1/j)) - k.
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EXAMPLE
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0.68538530275123375551874460543013077034359371013804...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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