Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 40

40
which is the condition for the convergence of the series
(71). It is therefore not allowable to conclude from
co —>
^>^anxn = oo that because the last serial
/2 — 0 /2 = 0
term on the right hand side of (76) may compensate the
first serial term and the result be a finite and well de-
fined y.
The determination of y in (73) is carried out entirely apart
from considerations concerning convergence and diverger.ee,
by converting (74) into a differential equatíon which operation
may be accomplished by the conversion formula, cf.
Soc. Sci. Isl., Divergent Power Series, Reykjavik, 1934, or
Serial Relations II, in Soc. Sci. Isl., Greinar I, 1940, pp. 177.
If we are able to integrate the differential equation, the
problem is wholly solved. However, in many cases we
have to resort to approximative methods, e. g. the asymp-
totic procedures set forth above, in order to discover the
nature of the function in question.
For those who are not acquainted with the conversion
formula, it is v,orth while to recall that the General Serial
Relation (3) may be applied with the same effect, if we
can find a linear relation between the coefíicients
an’ °n + V .....an + m‘
Let in (73) an — n + 1, then we have
(n + 2)an — (n + \)an + 1 = 0 (78)
Hence, with regard to (74) the right hand side of (3)
becomes zero, if q0 = 2x, qx — 3x — 1, q2 = 4x— 2,... ■
and consequently p0 = 2 x, px = 1 — x, p2 — p3 = p4... = 0 .
The left hand side of (3) is then:
2 x y (1 x) x ■— = 0 . (79)
whence by integration