27G
QUICK CONVERGING SERIES
TRACT 48.
1st, A
2 dly, A
3dly, a =
+
X ^ 3.4 + 5.4 :
I 1
* C 1 ~
O
Y x(l-
+ y x (1
3 9
1
~3A
I
3.49
I
TF
1
+++, b
5.9 s
1
FF
i
i
FT 3
i
tT 1
5.49 s
1
_ 5.9 S
1
7.4 3
1
7.49 3
J
' "T9 3
1
+ &c)
+ &C).
"f~ &c+ & c )-+ &c)
j + *W)-
3.49 ' 5.49 s 7.49
Now if each of these be transformed, by means of the dif-ferential series, in cor. 3 p. 64 of the late Mr. Simpson’sMathematical Dissertations, they will become of these verycommodious forms, viz,
1st, A = ‘
2dly, A =
3 dly, A =
+
Jl
To
3
To
4
T x f 1 +
7
x (1 + -
X (1 +
3.10
2
8a 12S , . ,
+ -t- -tttt; + & c )
3.10
4
"FTo
4
50 * ( l + 3.100
r +
h
5.104a5.108 a
7.10
+ T 7 ^ + &c) -12f
5.10 + T. 7 o + &c)
_6_
10
X (1 f
5.10
2
+
8a
5.100
4a
I-
12 £
7.1006£
+ &c).
50 X ^ + 3.50
5.10
4a
+ TVK +
7.10
6£
+ &c)
+ &c).
5.50 7.50
Where a, ?, y, &c, denote always the preceding terms ineach series.
Now it is evident that all these latter series are much easierthan the former ones, to which they respectively correspond;for, because of the powers of 10 here concerned, we havelittle more to do than to divide by the series of odd numbers1, 3, 5, 7, 9, &c.
Of all these three forms, the 2d is the fittest for comput*