114
(11.5)
II - Convergence: Discrete variables
l
b
bs+l - as+1
xSdx= - - - - -
a
8+1
for s E Z (in fact, 8 E ~) different from -1 and 0 < a < b (and in fact for
any a and b if 8 EN). The method above is due to Fermat who, curiously,
did not apply it to the case s = -1.
The concepts of integral and derivative are, in classical analysis, linked by
the "fundamental theorem of the integral calculus": if f is continuous, then
(i) every function F such that F' = f satisfies
(11.6)
l
X f(t)dt = F(x) - F(a)
for any x, (ii) if, conversely, one uses (6) to define a function F - no matter
what value one chooses for a -, then F' = f. Example 2 confirms point (ii);
Example 1 and point (ii) imply the formula log' x = l/x of nO 10, Theorem 3.
(To follow.)
12 - Series with positive terms
After this anticipation of Chap. V, we note several immediate applications of
Theorem 2 to series.
Theorem 4. For a series with positive terms to converge, it is necessary
and sufficient that its partial sums be bounded above. The sum of the series
is then the the least upper bound of the partial sums.
This is clear since they form an increasing sequence.
If for example one has two series E Un and E Vn with positive terms, if
the second converges, and if Un ::; Vn for all n (or only for all n sufficiently
large), then the first converges too, since its partial sums are smaller than
those of the second.
Theorem 4 shows also that if, in a series E u( n) with positive terms, one
regroups the terms
[u(l) + ... + U(Pl)] + [U(pl + 1) + ... + U(P2)] + ...
then the initial series and the new series E v(n) are simultaneously convergent or divergent. We already know that the convergence of the first series
implies that of the second without any hypothesis of positivity. To establish
the converse, one observes that the partial sum u(l) + ... + u(n) of the first
is majorised by the partial sum
v(l) + ... + v(n) = u(l) + ... + u(Pn)
of the second, since it contains more terms, all positive, than the first. So
if the partial sums of the second series are majorised, so similarly are those
(11.5)
II - Convergence: Discrete variables
l
b
bs+l - as+1
xSdx= - - - - -
a
8+1
for s E Z (in fact, 8 E ~) different from -1 and 0 < a < b (and in fact for
any a and b if 8 EN). The method above is due to Fermat who, curiously,
did not apply it to the case s = -1.
The concepts of integral and derivative are, in classical analysis, linked by
the "fundamental theorem of the integral calculus": if f is continuous, then
(i) every function F such that F' = f satisfies
(11.6)
l
X f(t)dt = F(x) - F(a)
for any x, (ii) if, conversely, one uses (6) to define a function F - no matter
what value one chooses for a -, then F' = f. Example 2 confirms point (ii);
Example 1 and point (ii) imply the formula log' x = l/x of nO 10, Theorem 3.
(To follow.)
12 - Series with positive terms
After this anticipation of Chap. V, we note several immediate applications of
Theorem 2 to series.
Theorem 4. For a series with positive terms to converge, it is necessary
and sufficient that its partial sums be bounded above. The sum of the series
is then the the least upper bound of the partial sums.
This is clear since they form an increasing sequence.
If for example one has two series E Un and E Vn with positive terms, if
the second converges, and if Un ::; Vn for all n (or only for all n sufficiently
large), then the first converges too, since its partial sums are smaller than
those of the second.
Theorem 4 shows also that if, in a series E u( n) with positive terms, one
regroups the terms
[u(l) + ... + U(Pl)] + [U(pl + 1) + ... + U(P2)] + ...
then the initial series and the new series E v(n) are simultaneously convergent or divergent. We already know that the convergence of the first series
implies that of the second without any hypothesis of positivity. To establish
the converse, one observes that the partial sum u(l) + ... + u(n) of the first
is majorised by the partial sum
v(l) + ... + v(n) = u(l) + ... + u(Pn)
of the second, since it contains more terms, all positive, than the first. So
if the partial sums of the second series are majorised, so similarly are those