A248946 -id:A248946 - cp's OEIS frontend

A248945 Decimal expansion of Sum_{n >= 1} (sin(1/n))^2.

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

Offset: 1

Clark Kimberling, Oct 18 2014

			1.32632440526665343354940377814268899683113371082571889555566414244...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Steve Chow, You did this wrong #Calc2Final, blackpenredpen video (2018)

Cf. A248946, 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, N[Sum[Sin[1/n]^2, {n, 1, 10000}], 120], yields only 7 correct decimals! *)

default(realprecision,150); 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

A248949 Decimal expansion of sum_{n >= 1} (sin(2/n))^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2014

			3.0136580258100591762101955819548270224075442567134903998883672964281449...

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

Cf. A248946, A248947, A248950.

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

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

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

A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 18 2021

Abel summation shows the series is convergent.

			0.96139431594573654724764595316154730686858269301058...

Konrad Knopp, Theory and Application of Infinite Series, Blackie, 1928, p. 313.
Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice C.3.7 2.3.b)4. p. 309.

Wikipedia, Summation by parts.

Cf. A070751, A070752, A096418, A248946.

nDgtsOutput:=110; nDgtsPrecision:=nDgtsOutput+10; SetDefaultRealField(RealField(nDgtsPrecision)); kMax:=Ceiling(1.395*nDgtsPrecision-3); mMax:=Ceiling(1.5*kMax); sum:=0.0; S1:=[0.0 : j in [1..kMax]]; n:=0; for m in [1..mMax] do S2:=S1; for k in [1..355] do n:=n+1; sum+:=Sin(Sin(n)/n); end for; S1[1]:=sum; for j in [1..kMax-1] do S1[j+1]:=(S2[j]+S1[j])/2; end for; end for; ChangePrecision(S1[#S1], nDgtsOutput); // The constants 1.395 and 1.5 were empirically derived; 355 is used because 355/Pi is very close to an odd integer. - Jon E. Schoenfield, Mar 21 2021

a(3)-a(104) from Jon E. Schoenfield, Mar 20 2021

A248950 Decimal expansion of sum_{n >= 1} sin(2/n^2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2014

			2.1763051477758324021862295698639364793494392031916981544951651550628667...

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

Cf. A248946, A248947, A248949.

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

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

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

A248951 Decimal expansion of sum_{n >= 1} (tan(2/n))^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2014

			9.0435286086582111526563827472958995791513335719871504139242512780719363...

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

Cf. A248946, A248947, A248948, A248950.

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

(* N[Sum[Tan[2/n]^2, {n, 1, Infinity}], 120], yields only 24 correct decimals *)

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

A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Apr 29 2023

Series Sum_{n>=1} sin(1/n) and Sum_{n>=1} tan(1/n) -> oo but with u(n) = (tan(1/n) - sin(1/n)), as u(n) ~ 1 / (2*n^3) when n -> oo, the series Sum_{n>=1} u(n) is convergent.

			Equals 0.822082200803588202935870118715993520730...

J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1 - 41.2, p. 47, Classes Préparatoires aux Grandes Ecoles, Ellipses, 1988.

Cf. A096444, A248946, A248948, A342680.

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

sumpos(n=1, tan(1/n) - sin(1/n)) \\ Michel Marcus, Apr 29 2023

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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A248945 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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A248949 Decimal expansion of sum_{n >= 1} (sin(2/n))^2.

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A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

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A248950 Decimal expansion of sum_{n >= 1} sin(2/n^2).

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Maple

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A248951 Decimal expansion of sum_{n >= 1} (tan(2/n))^2.

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A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).