A248945 -id:A248945 - cp's OEIS frontend

A248946 Decimal expansion of sum_{n >= 1} sin(1/n^2).

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

Offset: 1

Clark Kimberling, Oct 18 2014

			1.483522817309552286419299186361884222198966313715164026170885595688595...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A248945, A248947, A248949.

evalf(sum(sin(1/n^2), n=1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

(* N[Sum[Sin[1/n^2], {n, 1, Infinity}], 120], yields only 26 correct decimals *)

default(realprecision,120); sumpos(n=1,sin(1/n^2)) \\ Vaclav Kotesovec, Oct 20 2014

A248947 Decimal expansion of Sum_{n >= 1} tan(1/n)^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2014

			3.132799362818005226720034012858567638052280522418872401127192004180544...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A248945, A248946, A248948.

evalf(sum((tan(1/n))^2, n=1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

(* N[Sum[Tan[1/n]^2, {n, 1, Infinity}], 120], yields only 25 correct decimals *)

default(realprecision,120); sumpos(n=1,(tan(1/n))^2) \\ Vaclav Kotesovec, Oct 20 2014

A248948 Decimal expansion of Sum_{n >= 1} tan(1/n^2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2014

			2.208258813287175709084349680730347326245868074198770676727659915398266...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A248945, A248946, A248947.

evalf(sum(tan(1/n^2), n=1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

(* N[Sum[Tan[1/n^2], {n, 1, Infinity}], 120], yields only 25 correct decimals *)

default(realprecision,120); sumpos(n=1,tan(1/n^2)) \\ Vaclav Kotesovec, Oct 20 2014

A336405 Decimal expansion of Sum_{n>=1} log(n*sin(1/n)) (negated).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 20 2020

As v(n) = log(n*sin(1/n)) ~ -1/(6*n^2) when n -> oo, this series is convergent (zeta(2)/6 ~ 0.2741556778...).

			-0.28055633622915507960203968093919836217450282945971...

Cf. A013661, A233383, A248945, A295219, A336603.

Cf. A027641, A027642.

evalf(sum(log(n*sin(1/n)),n=1..infinity),50);

sumpos(n=1, log(n*sin(1/n))) \\ Michel Marcus, Jul 20 2020

Equals Sum_{n>=1} log(n*sin(1/n)).
Equals log(A295219).
From Amiram Eldar, Jul 30 2023: (Start)
Equals Sum_{k>=1} 2^(2*k-1)*(-1)^k*B(2*k)*zeta(2*k)/(k*(2*k)!), where B(k) is the k-th Bernoulli number.
Equals -Sum_{k>=1} zeta(2*k)^2/(k*Pi^(2*k)). (End)

More terms from Jinyuan Wang, Jul 21 2020

A362752 Decimal expansion of Sum_{k>=1} (1/k - sin(1/k)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 02 2023

			0.19189908550626482798114607722643984340430910237755...

Math Stackexchange, Closed Form for Sum_{k=1..n} sin(1/k), 2011.

Cf. A233383, A248945, A248946, A249022, A362753.

evalf(sum(1/k - sin(1/k), k = 1..infinity), 120);

sumalt(k = 1, (-1)^(k+1) * zeta(2*k+1)/(2*k+1)!)

Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k+1)/(2*k+1)!.

A362753 Decimal expansion of Sum_{k>=1} sin(1/k)/k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 02 2023

The value of the Hardy-Littlewood function H(x) = Sum_{k>=1} sin(x/k)/k at x = 1 (Hardy and Littlewood, 1936; Gautschi, 2004).

			1.47282823195618529629494738382314582532386592787930...

Walter Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, 2004. See Example 3.64, pp. 242-245.

Kara Garrison and Thomas E. Price, Approximating Sums of Infinite Series, 17th Biennial ACMS Conference MAY 27-30, 2009, Conference Proceedings (2009), pp. 74-83.
G. H. Hardy and J. E. Littlewood, Notes on the theory of series (xx): On Lambert series, Proc. London Math. Soc., Vol. s2-41, Issue 1 (1936), pp. 257-270.
Math Stackexchange, Does the series Sum_{k=1..n} sin(1/k)/k converge?, 2017.

Cf. A233383, A248945, A248946, A362752.

evalf(sum(sin(1/k)/k, k = 1 .. infinity), 120);

PARI
```
sumpos(k = 1, sin(1/k)/k)
```

Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k)/(2*k-1)!.

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A248946 Decimal expansion of sum_{n >= 1} sin(1/n^2).

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A248947 Decimal expansion of Sum_{n >= 1} tan(1/n)^2.

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A248948 Decimal expansion of Sum_{n >= 1} tan(1/n^2).

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A336405 Decimal expansion of Sum_{n>=1} log(n*sin(1/n)) (negated).

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A362752 Decimal expansion of Sum_{k>=1} (1/k - sin(1/k)).

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A362753 Decimal expansion of Sum_{k>=1} sin(1/k)/k.

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