A216548 -id:A216548 - cp's OEIS frontend

A013705 Decimal expansion of 4*Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

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

Offset: 1

N. J. A. Sloane

An approximation to Pi.
A case of "high precision fraud": curiously, among the first 40 digits, only 4 are wrong (in positions 7, 18, 19 and 30). - Jean-François Alcover, Apr 23 2013
This result arises because the sum is Pi - 2*10^-6 + 2*10^-18 - 10^-29 + 122*10^-42 - ... - Jon E. Schoenfield, Mar 11 2018
The constant is rational, as a finite product of rational numbers. The period of its decimal expansion is L = 1.7368897... * 10^33024, and so a(n + L) = a(n) for large enough n.

			3.1415906535897932404626433832695028841972913993751030509749446933498...

J. M. Borwein and P. B. Borwein, Strange series and high precision fraud, Amer. Math. Monthly 99, 7 (Aug. 1992), 622-640.
J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

4*sum(k=1, 500000, (-1.)^(k-1)/(2*k-1)) \\ Michel Marcus, Mar 11 2018

a(78)-a(80) corrected and more digits from Jon E. Schoenfield, Mar 11 2018

A013706 Decimal expansion of 2*Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

A deceptively correct-looking approximation to Pi/2.

			Pi/2:     1.570796326794896619231321691639751442098584699687552910487472...
This sum: 1.570786326794897619231321191639752052098583314687557962587445...
..........================^=================^^^======^^^^=====^=^^^===^^...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A013707, A195793.

RealDigits[2*Sum[(-1)^(k-1)/(2k-1),{k,50000}],10,130][[1]] (* Harvey P. Dale, Oct 23 2012 *)

Entry revised by N. J. A. Sloane, Sep 08 2012

A195793 Decimal expansion of arctan(1000000).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 24 2011

pi/2-arctan(1000000)<1/1000000; the first nonzero digits of pi/2-arctan(1000000) are as follows:

999999999999666666666666866666666666. The twelve 6's before 8 correspond to the limit shown at the end of the Mathematica program. What about the next eleven 6's?

			Let x=pi/2 and y=arc(1000000); then
x=1.57079632679489661923132169163975144209858469968755291048...
y=1.57079532679489661956465502497288477543191817587802910085...
x-y=0.000000099999999999966666666666686666666666652380963492...

Index entries for transcendental numbers

Cf. A000796, A195789.

For other approximations to Pi see A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548. - N. J. A. Sloane, Sep 08 2012

N[Pi/2, 100]
N[ArcTan[10^6], 100]
RealDigits[%]  (* A195793 *)
Limit[n^2 - (n^3) (Pi/2 - ArcTan[n]), n -> Infinity]
(* Limit equals 1/3 *)

atan(1e6) \\ Charles R Greathouse IV, Nov 20 2024

A216542 Decimal expansion of Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 08 2012

A deceptively correct-looking approximation to Pi/4. The constant is rational, by definition (a product of finitely many rational numbers).

			Pi/4:     0.785398163397448309615660845819875721049292349843776455...
This sum: 0.785393163397448809615660595819876026049291657343778981...
..........=================^========^^======^^=^=====^^^^^====^^^^...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

Digits:=300; M:=50000; add(evalf((-1)^(k-1)/(2*k-1)), k=1..M);

First[RealDigits[Sum[(-1)^(k-1)/(2*k-1), {k, 50000}], 10, 100]] (* Paolo Xausa, Apr 23 2024 *)

Equals A013706/2. - Hugo Pfoertner, Apr 23 2024

A216543 Decimal expansion of 4*Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 08 2012

An approximation to Pi. The constant is rational, by definition (a product of finitely many rational numbers).

			3.14157265358979523846264238327950410419716662937511592517489...

J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

First[RealDigits[4*Sum[(-1)^(k-1)/(2*k-1), {k, 50000}], 10, 100]] (* Paolo Xausa, May 11 2024 *)

A216544 Decimal expansion of Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 08 2012

An approximation to Pi/4. The constant is rational, by definition (a product of finitely many rational numbers).

			0.7853976633974483101156608458173757210493228498437757627437361733374541...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

Digits:=300; M:=500000; add(evalf((-1)^(k-1)/(2*k-1)), k=1..M);

RealDigits[Sum[(-1)^(k-1)/(2k-1),{k,500000}],10,130][[1]] (* Harvey P. Dale, Sep 11 2024 *)

A216545 Decimal expansion of 2*Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 08 2012

An approximation to Pi/2. The constant is rational, by definition (a product of finitely many rational numbers).

			1.57079532679489662023132169163475144209864569968755152548747...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A013707, A195793.

A216546 Decimal expansion of Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 08 2012

An approximation to Pi/4. The constant is rational, by definition (a product of finitely many rational numbers).

			0.7853981133974483096161608458198756960492923498468264552437...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A003881, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

Digits:=300; M:=5000000; add(evalf((-1)^(k-1)/(2*k-1)), k=1..M);

RealDigits[Total[Table[(-1)^(k-1)/(2k-1),{k,5*10^6}]],10,130][[1]] (* Harvey P. Dale, May 06 2015 *)

A216547 Decimal expansion of 2*Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 08 2012

An approximation to Pi/2. The constant is rational, by definition (a product of finitely many rational numbers).

			1.57079622679489661923232169163975139209858469969365291048747...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, in Pi: The Next Generation (2016) pp. 197-205.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A013707, A195793.

cp's OEIS Frontend

A013705 Decimal expansion of 4*Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

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A013706 Decimal expansion of 2*Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

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A195793 Decimal expansion of arctan(1000000).

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A216542 Decimal expansion of Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

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A216543 Decimal expansion of 4*Sum_{k=1..50000} (-1)^(k-1)/(2k-1).

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A216544 Decimal expansion of Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

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A216545 Decimal expansion of 2*Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

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A216546 Decimal expansion of Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

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