§l. Convergent sequences and series
69
the plane. Convergence of Un to U then means that this graph is "asymptotic"
to the horizontal line y = U in the plane (not by the definition of a limit -
one has no need to appeal to geometry for this - but, quite the contrary, by
the definition of an asymptote). A similar remark applies to the case of a
function f(x) defined for sufficiently large x E lR. and which tends to a limit
when x increases indefinitely: the fact that the function l/x tends to 0 at
infinity shows that its graph is asymptotic to the x axis.
When a sequence (un) tends to a limit u, one may write
lim Un = U or lim Un = U
n ..... oo
n ..... +oo
or even simply lim Un = u. Hardy's admonitions on the significance of the
symbol 00 apply here in full. Moreover, like the letter r in the last n O , the
letter n is only a phantom and one can replace it by any other sign, on
condition that one does so everywhere; one may very well write
lim u($) = u,
$ ..... 00
the mathematics will not change; however, it is forbidden to keep the first
sign $ and to replace the second by the sign £ since different letters a priori represent variables independent one of the other: one would then have
limu(£) = u(£), unless one specifies this by a relation such as £ = f($).
It is clear from the definition that for a sequence (un) to tend to a limit u
it is necessary and sufficient that the sequence with general term u - Un tends
to 0, which one can write in the form
Un = U + 0(1) when n ---+ +00
since the symbol 0(1) represents any sequence or function negligible with
respect to the constant function 1, i.e. tending to o.
Moreover, if Un = Vn +iwn is a sequence of complex terms, the inequalities
show that
lim(vn + iwn ) = v + iw -¢=} (lim Vn = v) & (lim Wn = w).
The inequality lIunl - lull ~ IUn - ul shows on the other hand that, for real
or complex sequences,
The converse is clearly false.
The most obvious example of a convergent sequence is obtained - and it
is a long way from being by chance - by starting with a real number u and
69
the plane. Convergence of Un to U then means that this graph is "asymptotic"
to the horizontal line y = U in the plane (not by the definition of a limit -
one has no need to appeal to geometry for this - but, quite the contrary, by
the definition of an asymptote). A similar remark applies to the case of a
function f(x) defined for sufficiently large x E lR. and which tends to a limit
when x increases indefinitely: the fact that the function l/x tends to 0 at
infinity shows that its graph is asymptotic to the x axis.
When a sequence (un) tends to a limit u, one may write
lim Un = U or lim Un = U
n ..... oo
n ..... +oo
or even simply lim Un = u. Hardy's admonitions on the significance of the
symbol 00 apply here in full. Moreover, like the letter r in the last n O , the
letter n is only a phantom and one can replace it by any other sign, on
condition that one does so everywhere; one may very well write
lim u($) = u,
$ ..... 00
the mathematics will not change; however, it is forbidden to keep the first
sign $ and to replace the second by the sign £ since different letters a priori represent variables independent one of the other: one would then have
limu(£) = u(£), unless one specifies this by a relation such as £ = f($).
It is clear from the definition that for a sequence (un) to tend to a limit u
it is necessary and sufficient that the sequence with general term u - Un tends
to 0, which one can write in the form
Un = U + 0(1) when n ---+ +00
since the symbol 0(1) represents any sequence or function negligible with
respect to the constant function 1, i.e. tending to o.
Moreover, if Un = Vn +iwn is a sequence of complex terms, the inequalities
show that
lim(vn + iwn ) = v + iw -¢=} (lim Vn = v) & (lim Wn = w).
The inequality lIunl - lull ~ IUn - ul shows on the other hand that, for real
or complex sequences,
The converse is clearly false.
The most obvious example of a convergent sequence is obtained - and it
is a long way from being by chance - by starting with a real number u and