Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 30

30
x
4
1
m + 1
m!
2! (m + 2)!
m!
1 0 0
m!
(m + 1)!
1 0.
m!
0
0
(/•+ 1)! (m + /•+ 1)! /'!(/// + /■)!
m!
(m + 1)!
= 0
For r=3 we obtain from (55), approximately:
(55)
x== + 4 \/(m + !) (m + 2) (ffi + 4)(ffi2 + 8ffi + 19)
v m4 + 17 m3 + 117 m2 + 379 m + 422
(56)
Hence for m = 0, x = + 2.40; for m = 1, x = + 3.79; for
m = 2, x = + 5.00 etc. According to Jahnke and Emde7),
the first value is correct to 2 decimal places while the
others should be 3,83 and 5,13 respectively. If we need
more accuracy in individual cases, we will find it more
practical to calculate the respective values directly from
(55) in that we choose r of adequate magnitude. Obviously
(55) may be written
1 0 2 1 t
X | m +T
p 1 Po 4 Pi Po
where p, and p0 are functions of r in that the ratio pjp0
converges towards a certain value with increasing r.
Certainly I need not say that the same procedure is
applicable to other power series with some restrictions to
be explained below. However, it is perhaps worth while
to point out that equations of higher degree may often be
solved by the same method.