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Sequences

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Sequences and Convergence

A #~{sequence} of real numbers is a function _ ~f #: &naturals. --> &reals. , _ written _ \{~a_~i\} _ where _ ~a_~i = ~f ( ~i ).

The sequence _ \{~a_~i\} _ is said to #~{converge} to ~a &in. &reals. , _ if , _ given any _ &epsilon. > 0 , _ &exist. ~n &in. &naturals. _ such that _ | ~a_~i - ~a | < &epsilon. _ &forall. ~i > ~n .

Write _ ~a_~i -> ~a , _ as _ ~i -> &infty. , _ or:

~a _ = _ lim{~a_~i,~i -> &infty.}

Convergence Results

Suppose _ ~a_~i -> ~a , _ and _ ~b_~i -> ~b , _ then

  1. ~a_~i + ~b_~i _ -> _ ~a + ~b
  2. ~a_~i - ~b_~i _ -> _ ~a - ~b
  3. ~a_~i ~x _ -> _ ~a~x , _ &forall. ~x &in. &reals.
  4. ~a_~i ~b_~i _ -> _ ~a~b
  5. ~a_~i / ~b_~i _ -> _ ~a / ~b

Proof:

  1. given any &epsilon. > 0 , _ &exist. ~n_1 ~n_2 _ such that _ | ~a_~i - ~a | < &epsilon. / 2 , _ &forall. ~i > ~n_1 , _ and _ | ~b_~i - ~b | < &epsilon. / 2, _ &forall. ~i > ~n_2.
    So _ | (~a_~i + ~b_~i) - (~a + ~b) | _ < _ &epsilon., _ &forall. ~i > max(~n_1,~n_2).
  2. - ~b_~i -> - ~b , _ since | - ~b_~i - ( - ~b) | = | ~b - ~b_~i | < &epsilon., &forall. ~i > ~n, and ~a_~i - ~b_~i = ~a_~i + ( - ~b_~i).
  3. ~a_~i~x -> ~a~x, for, given &epsilon. > 0, &exist. ~n such that | ~a_~i - ~a | < &epsilon. / | ~x | , _ &forall. ~i > ~n.
    | ~a_~i~x - ~a~x | = | ~a_~i - ~a | | ~x | < &epsilon. , _ &forall. ~i > ~n.
  4. from iii) ~a~b_~i -> ~a~b. Now | ~a_~i - ~a | | ~b_~i - ~b | = | ~a_~i~b_~i + ~a~b - ~a_~i~b - ~a~b_~i | = | ~a_~i~b_~i - ~a~b - (~a_~i~b - ~a~b) - (~a~b_~i - ~a~b) |
    >= | ~a_~i~b_~i - ~a~b | - | ~a_~i~b - ~a~b | - | ~a~b_~i - ~a~b |. - So | ~a_~i~b_~i - ~a~b | =< | ~a_~i - ~a | | ~b_~i - ~b | + | ~a_~i~b - ~a~b | + | ~a~b_~i - ~a~b |.
    Now given &epsilon. > 0, &exist. ~n_1, ~n_2 such that | ~a_~i - ~a | < &sqrt.(&epsilon. / 3), ~i > ~n_1; - | ~b_~i - ~b | < &sqrt.(&epsilon. / 3), ~i > ~n_2;
    and &exist. ~n_3, ~n_4 such that | ~a_~i~b - ~a~b | < &epsilon. / 3, ~i > ~n_3; - | ~a~b_~i - ~a~b | < &epsilon. / 3, ~i > ~n_4 (from iii);
    So | ~a_~i~b - ~a~b | =< &sqrt.(&epsilon. / 3) # &sqrt.(&epsilon. / 3) + &epsilon. / 3 + &epsilon. / 3 = &epsilon. , _ &forall. ~i > max\{~n_1,~n_2,~n_3,~n_4\}
  5. sufficient to show that 1 / ~b_~i -> 1 / ~b, (~b != 0) and then apply iv).
    &exist. ~n_1, such that | ~b_~i - ~b | < | ~b | / 2, - i.e. | ~b_~i | > | ~b | / 2, &forall. ~i > ~n_1 _ => _ | ~b_~i~b | > ~b^2 / 2 > 0.
    also given &epsilon. > 0, &exist. ~n_2, such that | ~b - ~b_~i | < ~b^2&epsilon. / 2, _ &forall. ~i > ~n_2
    | 1 / ~b_~i - 1 / ~b | = | ~b - ~b_~i | / | ~b ~b_~i | =< | ~b - ~b_~i | / (~b^2 / 2) =< (~b^2&epsilon. / 2) / (~b^2 / 2) _ = _ &epsilon., _ &forall. ~i > max\{~n_1,~n_2\}

Limits of Sequences

~{a_i} &rightarrow. ~a, ~{b_i} &rightarrow. ~b, _ if ~{a_i} &le. ~{b_i} &forall. ~i, _ then ~a &le. ~b.

Proof:
Assume ~a > ~b. Let &epsilon. = (~a - ~b) / 2 > 0 - &exist. ~n such that | ~{a_i} - ~a | < &epsilon. and | ~{b_i} - ~b | < &epsilon. &forall. ~i > ~n - i.e. - ~{a_i} > ~a - &epsilon. = ~b + &epsilon. > ~{b_i} _ &forall. ~i > ~n _ contradiction.

A sequence cannot converge to more than one limit.

Proof:
Suppose ~{a_i} &rightarrow. ~a and ~{a_i} &rightarrow. ~b we have ~{a_i} &le. ~{a_i} &forall. ~i, - so by above ~a &le. ~b and ~b &le. ~a _ => _ ~a = ~b.,

Monotone and Bounded Sequences

Increasing Sequences

A sequence is _ #{~{increasing}} _ if _ ~{a_i} =< ~a_{~i - 1} _ &forall. ~i. (#{~{strictly increasing}} if ~{a_i} < ~a_{~i - 1})

Decreasing Sequences

A sequence is _ #{~{decreasing}} _ if _ ~{a_i} >= ~a_{~i - 1} _ &forall. ~i. (#{~{strictly decreasing}} _ if _ ~{a_i} > ~a_{~i - 1})

In either case (increasing or decreasing) the sequence is said to be #{~{monotone}}

Bounded Above and Below

If &exist. ~b such that ~{a_i} =< ~b _ &forall. ~i , _ then the sequence is said to be #~{bounded above} by ~b.
~b is said to be an #~{upper bound} of the sequence.

If &exist. ~c such that ~{a_i} >= ~c _ &forall. ~i , _ then the sequence is said to be #~{bounded below} by ~c.
~c is said to be a #~{lower bound} of the sequence.

Axiom of Completeness of Real Numbers

An increasing sequence of real numbers which is bounded above converges to a real number.

[ Similarly (or equivalently) a decreasing sequence of real numbers which is bounded below converges to a real number. ]

#{Lemma}: _ if _ ~a_~i -> ~a , _ and ~a_~i is increasing _ then ~a_~i =< ~a , _ &forall. ~i

Proof:
Assume that ~a_~m > ~a for some ~m , _ then _ ~a_{~m + 1} >= ~a_~m , _ and by induction _ ~a_~i > ~a , _ &forall. ~i >= ~m . _ Now &exist. ~n such that _ | ~a_~i - ~a | < ~a_~m - ~a , _ &forall. ~i > ~n , _ in particular ~a_~i - ~a < ~a_~m - ~a , _ &forall. ~i > max&set. ~m, ~n &xset. _ => _ ~a_~i < ~a_~m , _ &forall. ~i > ~m , _ contradicting the fact that ~a_~i are increasing.

 

Nested Interval Theorem

Let &set.I_~k&xset. be a sequence of closed intervals, _ I_~k = [ ~a_~k, ~b_~k ] &subseteq. &reals., _ such that _ I_~k &contains. I_{~k + 1} , _ &forall. ~k , _ and length of I_~k -> 0 as ~k -> &infty..
Then &exist. unique real number &alpha. which lies in all the intervals, _ i.e. &set. &alpha. &xset. _ = _ &intersect._~k^{&infty.}_{= 1} _ I_~k .

Proof:
To be provided

 

 

Supremums and Infinums

Least Upper Bound

X &subseteq. &reals.. _ ~s is the #~{least upper bound} or #~{supremum} of X _ ( sup X ) _ if _ ~x =< ~s , _ &forall. ~x &in. S , _ and if ~x =< ~t , _ &forall. ~x &in. S _ then _ ~t = ~s.

Greatest Lower Bound

~q is the #~{greatest lower bound} or #~{infinum} of X _ ( inf X ) _ if _ ~x >= ~q , _ &forall. ~x &in. S , _ and if ~x >= ~r , _ &forall. ~x &in. S _ then _ ~r = ~q.

#{Lemma}: _ &set. ~a_~i &xset. increasing sequence, and ~a_~i -> ~a , _ then _ ~a _ = _ sup &set. ~a_~i &xset. . _ _ [ &set. ~a_~i &xset. decreasing _ => _ ~a _ = _ inf &set. ~a_~i &xset. . ]

( We write _ ~a_~i &uparrow. ~a _ for the increasing sequence, and _ ~a_~i &downarrow. ~a _ for the decreasing. )

#{Lemma}: _ A non-empty set of real numbers which is bounded above has a least integer upper bound _ (independent of Axiom of Completeness).

Proof:
To be provided

Axiom of Completeness (Alternative)

Any non-empty set _ X &subseteq. &reals. _ which is bounded above has a least upper bound (supremum).
[ Note this is equivalent to ''X bounded below has an infinum'' , _ - _ just take Y = &set. ~x | - ~x &in. X &xset. _ and apply axiom. ]

#{Lemma}: _ The two forms of the Axiom of Completeness are equivalent.

Proof:
To be provided

 

 

Divergence

A sequence _ &set. ~a_~i &xset. _ of real numbers which is not convergent is called #~{divergent}, _ there are three cases

  1. ~a_~i _ #~{diverges to + &infty.} _ (write _ ~a_~i -> + &infty.) _ if for any ~x &in. &reals. _ &exist. ~n &in. &naturals. _ ~a_~i > ~x , _ &forall. ~i > ~n .
  2. ~a_~i _ #~{diverges to - &infty.} _ (write _ ~a_~i -> - &infty.) _ if for any ~x &in. &reals. _ &exist. ~n &in. &naturals. _ ~a_~i < ~x , _ &forall. ~i > ~n .
  3. ~a_~i _ #~{oscillates} _ if it does not diverge to +- &infty..

#{Example}: _ ~a_~i = ~x^~i , _ some ~x &in. &reals.

  1. ~x > 1, _ i.e. _ ~x = 1 + ~y , _ some ~y > 0. _ &therefore. ~x^~i = ( 1 + ~y )_~i > ~i~y . _ By Archimedes ~i~y can be as large as we like _ so _ ~x_~i -> + &infty. .
  2. ~x = 1, _ ~x^~i = 1 _ &forall. ~i _ so _ ~x_~i -> 1.
  3. 0 =< ~x < 1, _ say ~x = 1 / ~z , _ where _ ~z > 1, _ then given &epsilon. > 0 _ &exist. ~n such that _ ~z^~i > 1 / &epsilon. _ &forall. ~i > ~n, _ so _ ~x_~i -> 0 .
  4. -1 < ~x < 0, _ | ~x | ^~i -> 0 _ (as in 3) _ &therefore. by definition _ ~x^~i -> 0.
  5. ~x = -1, _ ~x^~i = 1, ~i even, _ ~x^~i = -1, ~i odd, _ &therefore. _ ~x^~i oscillates.
  6. ~x < -1, _ | ~x | ^~i -> &infty., _ even terms positive, odd terms negative, _ &therefore. _ &set. ~x^~i &xset. oscillating but unbounded.

#{Lemma}: _ An increasing sequence which is unbounded diverges to + &infty..

Proof:
~x &in. &reals., _ if there is no ~n such that _ ~a_~n > ~x , _ then &set. ~a_~i &xset. bounded by ~x _ - _ contradiction. _ So _ &exist. ~n such that _ ~a_~n > ~x _ and since ~a_~i increasing _ ~a_~i > ~x , _ &forall. ~i > ~n.

Bounded Sequences

Let &set. ~a_~i &xset. be a sequence (not necessarily monotone). _ &set. ~a_~i &xset. is _ #~{bounded} _ if &exist. ~k &in. &reals. _ such that _ | ~a_~i | =< | ~k | _ &forall. ~i.

Lim inf and Lim sup

Let _ ~u_~n = sup_{~i >= ~n} &set. ~a_~i &xset. _ and _ ~l_~n = inf_{~i >= ~n} &set. ~a_~i &xset. _ then

  1. ~l_~n =< ~a_~n =< ~u_~n _ &forall. ~n
  2. &set. ~u_~n &xset. is decreasing _ and _ &set. ~l_~n &xset. is increasing
  3. &set. ~u_~n &xset. bounded below (by ~l_1 say) _ and _ &set. ~l_~n &xset. bounded below (by ~u_1 say) _ so
    _ _ _ ~u_~n -> ~u _ say, _ _ ~u _ is called _ #~{lim sup} &set. ~a_~i &xset.
    _ _ _ ~l_~n -> ~l _ say, _ _ ~l _ is called _ #~{lim inf} &set. ~a_~i &xset.

#{Lemma}: _ &set. ~a_~i &xset. bounded, and ~a_~i -> ~a _ <=> _ lim sup &set. ~a_~i &xset. = lim inf &set. ~a_~i &xset. = ~a

Proof:
To be provided

Cauchy Sequence

A sequence &set. ~a_~i &xset. is said to be a _ #~{Cauchy sequence} _ if given &epsilon. > 0 _ &exist. ~n _ such that _ | ~a_~i - ~a_~j | < &epsilon. _ &forall. ~i, ~j > ~n.

 

 

 

Cauchy's General Principle of Convergence

(For the real numbers)

A sequence &set. ~a_~i &xset._{~i &in. &naturals.} &subset. &reals. is convegent _ <=> _ it is a Cauchy sequence

Proof:
To be provided

Convergence Tests

Let &set. ~a_~i &xset. be a sequence of positive real numbers.

  1. ~a_{~i + 1} / ~a_~i _ =< _ ~k < 1 _ => _ ~a_~i -> 0
  2. ~a_{~i + 1} / ~a_~i _ -> _ ~k < 1 _ => _ ~a_~i -> 0
  3. ~a_{~i + 1} / ~a_~i _ >= _ ~l > 1 _ => _ ~a_~i -> &infty.
  4. ~a_{~i + 1} / ~a_~i _ -> _ ~l > 1 _ => _ ~a_~i -> &infty.

Proof:
To be provided

#{Examples}:

1) _ ~a_~i _ = _ ~i ^~k / ~x^~i

fract{~a_{~i + 1}, ~a_~i} _ = _ fract{( ~i + 1 ) ^~k ~x^~i , ~i ^~k ~x^{~i + 1}} _ = _ script{rndb{1 + fract{1, ~i }},,,~k,} fract{1,~x}

for fixed ~k : _ _ script{rndb{1 + fract{1, ~i }},,,~k,} _ -> _ 1 , _ as ~i -> &infty.

so _ _ fract{~i ^~k, ~x^~i} _ -> _ fract{1, ~x} , _ as ~i -> &infty.

2) _ ~a_~i _ = _ ~x^~i / ~i#!

fract{~a_{~i + 1}, ~a_~i} _ = _ fract{~i#! ~x^{~i + 1}, ~x^~i (~i + 1)#!} _ = _ fract{~x, ~i + 1} _ -> _ 0

3) _ ~a_~i = ~i#! / ~i^~i

fract{~a_{~i + 1}, ~a_~i} _ = _ fract{( ~i + 1 )#! ~i^~i,( ~i + 1 )^{~i + 1} ~i#!} _ = _ fract{( ~i + 1 ) ~i^~i,( ~i + 1 )^~i # ( ~i + 1 ) }

_ = _ fract{~i^~i,( ~i + 1 )^~i} _ = _ fract{1,( 1 + ( 1 / ~i ) )^~i}

but

script{rndb{1 + fract{ 1 , ~i }},,,~i,} _ _ = _ _ 1 + matrix{~i/1}fract{1,~i} _ _ _ + _ _ _ matrix{~i/2}fract{1,~i^2} + ... + matrix{~i/~i}fract{1,~i^~i} _ _ _ >= _ _ _ 2

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( _ _ = _ 2 _ _ ) _ _ _ _ _ _ _ ( _ _ _ _ _ _ _ > _ 0 _ _ _ _ _ _ _ )

So

fract{~a_{~i + 1}, ~a_~i} _ _ =< _ _ fract{1,2} , _ _ &therefore. _ ~a_~i -> 0 _ as _ ~i -> &infty.

Divergent Sequences Examples

  1. ~k _ fixed integer, _ ~k > 0, _ ~a_~i = ~i ^~k _ -> &infty. _ as ~i -> &infty.
  2. ~x _ fixed integer, _ ~x > 0, _ ~a_~i = ~x^~i _ -> &infty. _ as ~i -> &infty.
  3. ~a_~i = ~i #! _ -> &infty. _ as ~i -> &infty.
  4. ~a_~i = ~i ^~i _ -> &infty. _ as ~i -> &infty.