On a new type of rational and highly convergent series, by which the ratio of the circumference to the diameter is able to be expressed

5 Pages·1779·0.061 MB·English

by  Leonhard Euler

On a new type of rational and highly
5 convergent series, by which the ratio of the
0
0
2 circumference to the diameter is able to be

g
u expressed∗
A

9

Leonhard Euler†
1
v

0
7
1
1. The principle, from which these series are deduced, rests in this bi-
8
0 nomial formula: 4 + x4, which is evidently composed of these two rational
5
factors: 2 + 2x + xx and 2 − 2x + xx. Then indeed it at once follows for
0
/ this integral formula: ∂x(2+2x+xx), which we shall indicate with the sign ⊙,
O 4+x4
to be reduced to this:R⊙ = ∂x , whose integral, having been obtained
H 2 2x+xx
so that it vanishes when it iRs p−ut x = 0, is Atang. x . Whereby it may be
h. 2 x
observed in the case x = 1 to be ⊙ = π; while inde−ed in the case x = 1 it
t
a 4 2
will be ⊙ = Atang. 1; then indeed in the case x = 1 it will be ⊙ = Atang. 1.
m
3 4 7
It is noted moreover for it to be
:
v
i 1 1 π
X 2Atang. +Atang. = Atang.1 = .
r 3 7 4
a
∗Delivered to the St.–Petersburg Academy June 17, 1779. Originally published as De
novo genere serierum rationalium et valde convergentium, quibus ratio peripheriae ad di-
ametrum exprimi potest, Nova Acta Academiae Scientarum Imperialis Petropolitinae 11
(1798),150–154,andrepublishedinLeonhardEuler,OperaOmnia,Series1: Operamathe-
matica,Volume16,Birkh¨auser,1992. Acopyoftheoriginaltextisavailableelectronically
attheEulerArchive,athttp://www.eulerarchive.org. ThispaperisE706intheEnestr¨om
index.
†Date of translation: August 8, 2005. Translated from the Latin by Jordan Bell,
3rd year undergraduate in Honours Mathematics, School of Mathematics and Statistics,
Carleton University, Ottawa, Ontario, Canada. Email: [email protected]. This
translation was written during an NSERC USRA supervised by Dr. B. Stevens.
1
2. With therefore this integral formula indicated by the sign ⊙ which is
comprised by three parts, each of which we shall unfold separately, which by
the grace of brevity we shall indicate by the following characters:
∂x x∂x xx∂x
I. = Y; II. = X; III. = ♂;
Z 4+x4 Z 4+x4 Z 4+x4
so that it will thus be
x
⊙ = 2Y+2X+♂ = Atang. .
2−x
Now therefore we may unfold these three integral formulas in the usual man-
ner into infinite series, which are thereupon to be formed, insofar as it will
be
1 1 x4 x8 x12 x16
= 1− + − + − etc. .
4+x4 4(cid:16) 4 42 43 44 (cid:17)
3. But if now first we adjoin this series with ∂x and we then integrate,
the first formula Y will be expressed by the following series:
x 1 x4 1 x4 2 1 x4 3
Y = [1− · + − + etc.].
4 5 4 9(cid:16) 4 (cid:17) 3(cid:16) 4 (cid:17)
While indeed adjoining the former series by x∂x and integrating gives
xx 1 x4 1 x4 2 1 x4 3
X = [1− · + − + etc.].
8 3 4 5(cid:16) 4 (cid:17) 7(cid:16) 4 (cid:17)
Then adjoining the very same series with xx∂x and integrating produces
x3 1 1x4 1 x4 2 1 x4 3
♂ = [ − + − + etc.].
4 3 7 4 11(cid:16) 4 (cid:17) 15(cid:16) 4 (cid:17)
4. With therefore it being ⊙ = 2Y + 2X + ♂, we shall unfold some
particular cases recalled from before, in which it is x = 1, x = 1 and x = 1,
2 4
of which the first is x4 = 1; for the second indeed it is x4 = 1 ; for the
4 4 4 64
third indeed x4 = 1 ; from which it stands open for the two last cases to
4 1024
converge most greatly, but that the first, whose terms decrease by a ratio of
four, indeed converges more so than the series of Leibnitz, by taking an arc
whosetangentis 1 ,seeing thatthiscalculationisperturbedbynoirrational.
√3
2
The expansion of the first case,
where x = 1 and ⊙ = Atang. π.
4
5. Seeing therefore here that it is x4 = 1, our three principal series for
4 4
Y,X,⊙ [sic] proceed in the following way:
Y = 1[1− 1 · 1 + 1(1)2 − 1 (1)3 + 1 (1)4 − etc.]
4 5 4 9 4 13 4 17 4
X = 1[1− 1 · 1 + 1(1)2 − 1(1)3 + 1(1)4 − etc.]
8 3 4 5 4 7 4 9 4
♂ = 1[1 − 1 · 1 + 1 (1)2 − 1 (1)3 + 1 (1)4 − etc.]
4 3 7 4 11 4 13 4 19 4
6. Seeing therefore that it is ⊙ = 2Y+2X+♂ = π, by multiplying the
4
value of π by 4, the following three series are expressed
2(1− 1 · 1 + 1 · 1 − 1 · 1 + 1 · 1 − etc.)
5 4 9 42 13 43 17 44

π = 1(1− 1 · 1 + 1 · 1 − 1 · 1 + 1 · 1 − etc.)
 3 4 5 42 7 43 9 44
1(1 − 1 · 1 + 1 · 1 − 1 · 1 + 1 · 1 − etc.)
3 7 4 11 42 15 43 19 44


7. From these particular three series the ratio of the circumference to the
diameter is able to be calculated with much less work than by the series of
Leibnitz, which method the most meritous Authors Sharp, Machin and de
Lagny have used, of whom the first has determined π in a decimal fraction
to 72 figures, the second to 100, and the last indeed to 128. And truly they
were able to lift up the following cases with much more effort.
The expansion of the second case,
where x = 1.
2
8. In this case it will therefore be x4 = 1 , from which the three series
4 64
are drawn forth in the following way:
Y = 1(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
8 5 64 9 642 13 643
X = 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
32 3 64 5 642 7 643
♂ = 1 (1 − 1 · 1 + 1 · 1 − 1 · 1 + etc.)
32 3 7 64 11 642 15 643
3
9. Therefore with it 2Y+2X+♂ = Atang. 1, it will be
3
1(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
1 4 5 64 9 642 13 643

Atang. = 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
3 16 3 64 5 642 7 643
1 (1 − 1 · 1 + 1 · 1 − 1 · 1 + etc.)
32 3 7 64 11 642 15 643


Even though here these three series are to be computed, however, because
each successively decreases by the same ratio 1 : 64, this labor will be able
to be shortened in a wonderful way.
The expansion of the third case,
where x = 1.
4
10. Seeing therefore here that it is x4 = 1 , our three principal series
4 1024
will be had as follows:
Y = 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
16 5 1024 9 10242 13 10243
X = 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
128 3 1024 5 10242 7 10243
♂ = 1 (1 − 1 · 1 + 1 · 1 − 1 · 1 + etc.)
256 3 7 1024 11 10242 15 10243
11. Therefore with 2Y+2X+♂ = Atang. 1, it will properly be by joining
7
these series:
1(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
1 8 5 1024 9 10242 13 10243

Atang. = 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
7 64 3 1024 5 10242 7 10243
1 (1 − 1 · 1 + 1 · 1 − 1 · 1 + etc.)
256 3 7 1024 11 10242 15 10243


An application of the last two cases for expressing a
circumference by highly convergent series.
12. With, as we have already observed, π = 2Atang. 1 +Atang. 1 it will
4 3 7
be π = 8Atang. 1 +4Atang. 1, by substituting the value of π into the series
3 7
4
found above, the following six series may be expressed together:
2(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
5 64 9 642 13 643
1(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
2 3 64 5 642 7 643

1(1 − 1 · 1 + 1 · 1 − 1 · 1 + etc.)
π = 4 3 7 64 11 642 15 643
1(1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
2 5 1024 9 10242 13 10243
1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
16 3 1024 5 10242 7 10243

 1 (1− 1 · 1 + 1 · 1 − 1 · 1 + etc.)
64 7 1024 11 10242 15 10243


This occurs as most noteworthy, because all these series proceed only by
powers of two.
5