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A323736 Decimal expansion of the limit of (Sum_{j=1..k} arcsin(1/j)) - log(k) as k approaches infinity. 2
1, 1, 8, 4, 9, 0, 5, 2, 0, 2, 9, 4, 8, 6, 4, 5, 0, 2, 2, 7, 3, 9, 3, 5, 7, 1, 2, 6, 3, 8, 3, 2, 0, 2, 4, 1, 2, 6, 7, 8, 0, 2, 2, 5, 1, 7, 7, 6, 4, 9, 6, 9, 0, 1, 5, 1, 4, 1, 5, 2, 2, 4, 3, 5, 7, 5, 1, 0, 5, 3, 0, 0, 0, 7, 4, 3, 7, 4, 0, 3, 7, 4, 0, 3, 1, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Draw a set of circular arcs, with each arc having its center at the origin and drawn counterclockwise so that it connects the line x=1 to the 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), which approaches 1/k as k increases.

Let s(k) = Sum_{j=1..k} alpha(j) = Sum_{j=1..k} arcsin(1/j); since arcsin(1/j) > 1/j for j >= 1, and Sum_{j=1..k} 1/j diverges as k increases, s(k) must likewise diverge as k increases.

Lim_{k->infinity} ((Sum_{j=1..k} 1/j) - log(k)) approaches the Euler-Mascheroni constant, 0.5772156649015... (cf. A001620); this sequence gives the decimal expansion of lim_{k->infinity} (s(k) - log(k)).

(As k increases, the length of the k-th arc approaches 1, and (sum of the lengths of the first k arcs) - k also approaches a constant; see A323737.)

LINKS

Table of n, a(n) for n=1..87.

FORMULA

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

EXAMPLE

1.18490520294864502273935712638320241267802251776496...

CROSSREFS

Cf. A001620, A323737.

Sequence in context: A124012 A000803 A198063 * A093208 A155064 A011225

Adjacent sequences:  A323732 A323733 A323734 * A323737 A323738 A323739

KEYWORD

nonn,cons

AUTHOR

Jon E. Schoenfield, Feb 03 2019

STATUS

approved

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Last modified September 19 04:39 EDT 2019. Contains 327187 sequences. (Running on oeis4.)