§2. Series expansions
391
a power series in y-l converging for y > 1; on passing to the series of primitives as if dealing with a power series in y we obtain the relation
' " 1.3 ... (2p - 1) 2 p /
argcoshy = logy + C - ~
y2p
2.4 .... 2p
p~l
where C is a constant. This formal calculation, "integrating term-by-term"
a series in y-l and not in y, is justified (while we do not yet have Chap. V
at our disposal) by checking that if one differentiates the series obtained
term-by-term, which produces the initial series (4), we obtain a series which
converges normally on every compact interval K eJI, +oo[ (Chap. III, § 6.
nO 17, Corollary of Theorem 19). Indeed, since the radius of convergence of
Newton's series is equal to 1, the series (4) converges normally on the interval
[a, +oo[ for any a> 1 (Chap. III, § 4, n° 17, example 1), a more than sufficient
result.
It remains to calculate the constant C. For this we use the relation
argcoshy = log(y + (y2 - I)!) = logy + log(I + (1 _ y-2)!)
below, and lets y tend to +00; the last term tends to log 2, whence
argcoshy=logy+log2+0(I) as y~+oo;
But when y ~ +00, i.e. when y-l tends to 0, the sum of the power series in
y-l obtained above tends to its constant term, namely o. Hence we find that
argcoshy = logy + C + 0(1), whence C = log 2.
The formula
(16.5)
' " 1.3 ... (2p - 1) 2p /
arg cosh y = logy + log 2 - ~
4
y2p
2 .... . 2p
p~l
yields an asymptotic expansion (Chap. VI) of the function when y is very
large.
As y tends to 1 the general term up(y) of the series is positive and tends
to up(I) remaining::; up(I). If one shows that EUp(I) converges one can
then pass to the limit under the E sign (Chap. III, nO 8, Theorem 9). Now
up+1(I)/up(I) = (2p + I)2p/(2p + 2)2 = (1 + I/2p)(1 + I/p)-2
whence
(16.6)
with s = 3/2 > 1. Now (Vol. II, Chap. VI, n° 5: Gauss' criterion, easy to
prove) any series of positive terms satisfying a relation (16.6) is convergent
391
a power series in y-l converging for y > 1; on passing to the series of primitives as if dealing with a power series in y we obtain the relation
' " 1.3 ... (2p - 1) 2 p /
argcoshy = logy + C - ~
y2p
2.4 .... 2p
p~l
where C is a constant. This formal calculation, "integrating term-by-term"
a series in y-l and not in y, is justified (while we do not yet have Chap. V
at our disposal) by checking that if one differentiates the series obtained
term-by-term, which produces the initial series (4), we obtain a series which
converges normally on every compact interval K eJI, +oo[ (Chap. III, § 6.
nO 17, Corollary of Theorem 19). Indeed, since the radius of convergence of
Newton's series is equal to 1, the series (4) converges normally on the interval
[a, +oo[ for any a> 1 (Chap. III, § 4, n° 17, example 1), a more than sufficient
result.
It remains to calculate the constant C. For this we use the relation
argcoshy = log(y + (y2 - I)!) = logy + log(I + (1 _ y-2)!)
below, and lets y tend to +00; the last term tends to log 2, whence
argcoshy=logy+log2+0(I) as y~+oo;
But when y ~ +00, i.e. when y-l tends to 0, the sum of the power series in
y-l obtained above tends to its constant term, namely o. Hence we find that
argcoshy = logy + C + 0(1), whence C = log 2.
The formula
(16.5)
' " 1.3 ... (2p - 1) 2p /
arg cosh y = logy + log 2 - ~
4
y2p
2 .... . 2p
p~l
yields an asymptotic expansion (Chap. VI) of the function when y is very
large.
As y tends to 1 the general term up(y) of the series is positive and tends
to up(I) remaining::; up(I). If one shows that EUp(I) converges one can
then pass to the limit under the E sign (Chap. III, nO 8, Theorem 9). Now
up+1(I)/up(I) = (2p + I)2p/(2p + 2)2 = (1 + I/2p)(1 + I/p)-2
whence
(16.6)
with s = 3/2 > 1. Now (Vol. II, Chap. VI, n° 5: Gauss' criterion, easy to
prove) any series of positive terms satisfying a relation (16.6) is convergent