A013705 -id:A013705 - cp's OEIS frontend

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

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.

A216548 Decimal expansion of 4*Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

Original entry on oeis.org

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

			3.1415924535897932384646433832795027841971693993873058209749418223...

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.

A013707 Decimal expansion of Sum_{k=1..50000} (-1)^(k+1) / k.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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. A013705, A013706.

RealDigits[Sum[(-1)^(k+1)/k,{k,50000}],10,120][[1]] (* Harvey P. Dale, Jul 27 2017 *)

log(2) + (Psi(25000 + 1/2) - Psi(25001))/2. - Robert Israel, Feb 03 2015

Corrected and extended by Jon E. Schoenfield, Feb 03 2015

cp's OEIS Frontend

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

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