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II - Convergence: Discrete variables
17 - Infinite limits
When a sequence of numbers Un does not converge, it may happen that as n
increases indefinitely, the Un vary in an irregular way (the case of the sequence
sin n for example), or that they increase, or decrease, indefinitely, in other
words "tend to +00 or to -00". The aim of this nO is to study this case
summarily - experience is more helpful than theorems in situations of this
type - showing which algebraic operations on sequences lead to predictable
results. We shall consider only sequences of real numbers in what follows.
First, we shall write
limun = +00
when, for any number A, one has Un > A for n sufficiently large; the relation
limun =-00
has a similar meaning, except that one has Un < A for n sufficiently large.
Obviously one tends to choose "very large" positive numbers for A in the
first case (or very large negative numbers in the second) since if one can
check that Un > 10100000 for n sufficiently large, there is no point in taking
the trouble to check that Un > _10 123 too ... But, strictly speaking, the
definition imposes no hypothesis on A.
Consider for example an increasing sequence (un); to say that, for any A,
one has Un > A for n large is clearly equivalent to saying that the sequence is
not majorised, in other words that it diverges. One might therefore say that
every increasing sequence converges, possibly to +00.
This remark also allows one to write
LUn < +00
to express the fact that a series with positive terms converges and to attribute
to it the sum +00 in the opposite case; one can similarly express absolute
convergence by writing that
These are pure conventions of language or of expression and are not to be
confused with theorems; reread Hardy, end of n° 2. One can apply them to
unordered series LiEf u(i).
The rules of calculus that apply to finite limits extend to a certain extent
to infinite limits too.
(I) If lim Un = +00 and Vn 2: Un for n large, then lim Vn = +00. Evident.
(II) If lim Un = +00, then lim c. Un = +00 or -00 as c is > 0 or < O. Evident.
(III) The relation lim Un = +00 implies lim l/un = O. Evident. The converse
is true if Un > 0 for n large. This comes from the fact that, if Un is > 0
for n large, the relation l/un < r is equivalent to Un > l/r.
II - Convergence: Discrete variables
17 - Infinite limits
When a sequence of numbers Un does not converge, it may happen that as n
increases indefinitely, the Un vary in an irregular way (the case of the sequence
sin n for example), or that they increase, or decrease, indefinitely, in other
words "tend to +00 or to -00". The aim of this nO is to study this case
summarily - experience is more helpful than theorems in situations of this
type - showing which algebraic operations on sequences lead to predictable
results. We shall consider only sequences of real numbers in what follows.
First, we shall write
limun = +00
when, for any number A, one has Un > A for n sufficiently large; the relation
limun =-00
has a similar meaning, except that one has Un < A for n sufficiently large.
Obviously one tends to choose "very large" positive numbers for A in the
first case (or very large negative numbers in the second) since if one can
check that Un > 10100000 for n sufficiently large, there is no point in taking
the trouble to check that Un > _10 123 too ... But, strictly speaking, the
definition imposes no hypothesis on A.
Consider for example an increasing sequence (un); to say that, for any A,
one has Un > A for n large is clearly equivalent to saying that the sequence is
not majorised, in other words that it diverges. One might therefore say that
every increasing sequence converges, possibly to +00.
This remark also allows one to write
LUn < +00
to express the fact that a series with positive terms converges and to attribute
to it the sum +00 in the opposite case; one can similarly express absolute
convergence by writing that
These are pure conventions of language or of expression and are not to be
confused with theorems; reread Hardy, end of n° 2. One can apply them to
unordered series LiEf u(i).
The rules of calculus that apply to finite limits extend to a certain extent
to infinite limits too.
(I) If lim Un = +00 and Vn 2: Un for n large, then lim Vn = +00. Evident.
(II) If lim Un = +00, then lim c. Un = +00 or -00 as c is > 0 or < O. Evident.
(III) The relation lim Un = +00 implies lim l/un = O. Evident. The converse
is true if Un > 0 for n large. This comes from the fact that, if Un is > 0
for n large, the relation l/un < r is equivalent to Un > l/r.