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Series

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Series

A _ #~{series} _ is the sum of a sequence, _ &set. ~a_~i &xset., _ of real numbers

sum{~a_~i,~i = 0, &infty.} _ = _ ~a_1 + ~a_2 + ...

The ~n^{th} _ #~{partial sum} _ of the series is:

~s_~n _ = _ sum{~a_~i,~i = 0, ~n} _ = _ ~a_1 + ~a_2 + ... + ~a_~n

the series, _ &sum._1^{&infty.} ~a_~i _ is said to be _ #~{convergent} _ if

~s_~n _ -> _ ~s _ _ _ _ as _ _ _ _ ~n _ -> _ &infty.

Sums of Series

#{Lemma}:

sum{~a_~i,~i = 0,&infty.} _ -> _ ~s , _ sum{~b_~i,~i = 0,&infty.} _ -> _ ~t _ _ _ => _ _ _ sum{~a_~i +- ~b_~i,~i = 0,&infty.} _ -> _ ~s +- ~t

Proof: _ ~s_~n +- ~t_~n = &sum. ( ~a_~i +- ~b_~i ), _ ~s_~n -> ~s, _ ~t_~n -> ~t, _ so _ ~s_~n +- ~t_~n -> ~s +- ~t.

#{Lemma}: _ If the series _ &sum._~i ~a_~i _ is convergent then _ ~a_~i -> 0 _ as ~i -> &infty.

Proof: _ ~s_~n -> ~s, _ also _ ~s_{~n+1} -> ~s, _ so _ ~a_~n = ~s_{~n+1} - ~s_~n _ -> _ 0.

Examples of Series

Examples of series are shown on a separate page. These include geometric , exponential , harmonic , and alternate harmonic series.

Convergence Tests for Series

#{Lemma}: _ ~a_~n >= 0 &forall. ~n , _ &sum._1^{&infty.} _ ~a_~n _ convergent _ <=> _ the partial sums, _ ~s_~r _ are bounded.

Proof: _ ~s_~r is increasing and bounded _ => _ convergence
Conversely _ ~s_~r is increasing and ~s_~r -> ~s _ then _ ~s_~r is bounded by ~s.

The convergence tests follow immediately from the above result:

Comparison Test

( ~a_~n , ~b_~n >= 0 _ &forall. ~n. )

~a_~n / ~b_~n =< 1 _ &forall. ~n _ _ then _ &sum._1^{&infty.} _ ~b_~n _ convergent _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
~a_~n / ~b_~n >= 1 _ &forall. ~n _ _ then _ &sum._1^{&infty.} _ ~b_~n _ divergent _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.

Comparison Test (Limit Form)

( ~a_~n , ~b_~n >= 0 _ &forall. ~n. )

~a_~n / ~b_~n -> ~k > 0 _ &forall. ~n   &sum._1^{&infty.} _ ~b_~n _ convergent _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
&sum._1^{&infty.} _ ~b_~n _ divergent _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.

D'Alembert's Ratio Test

( ~a_~n > 0 _ &forall. ~n. )

~a_{~n+1} / ~a_~n =< ~k < 1 _ &forall. ~n > ~n_0 _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
~a_{~n+1} / ~a_~n _ >= _ 1 _ _ &forall. ~n > ~n_0 _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.

D'Alembert's Ratio Test (Limit Form)

( ~a_~n > 0 _ &forall. ~n. )

~a_{~n+1} / ~a_~n _ -> _ ~k   ~k < 1 _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
~k > 1 _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.
~k = 1 _ _ _ _ inconclusive.

Cauchy's Root Test

( ~a_~n >= 0 _ &forall. ~n. )

~a_~n^{1/~n} =< ~k < 1 _ &forall. ~n > ~n_0 _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
~a_~n^{1/~n} _ >= _ 1 _ _ &forall. ~n > ~n_0 _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.

Cauchy's Root Test (Limit Form)

( ~a_~n >= 0 _ &forall. ~n. )

~a_~n^{1/~n} _ -> _ ~k   ~k < 1 _ => _ &sum._1^{&infty.} _ ~a_~n _ convergent.
~k > 1 _ => _ &sum._1^{&infty.} _ ~a_~n _ divergent.
~k = 1 _ _ _ _ inconclusive.

Convergence Tests Examples

1) _ _ _ _ _ _ ~a_~n _ = _ fract{~n,10^6~n^2 + (10)^{10^6}} , _ _ _ _ _ _ compare with _ ~b_~n = fract{1,~n}

fract{~a_~n,~b_~n} _ = _ fract{1,10^6 + fract{(10)^{10^6},~n^2}} _ _ _ _ --> _ _ _ _ 10^6

sum{~b_~n,1,&infty.} divergent _ _ _ _ _ _ => _ _ _ _ _ _ sum{~a_~n,1,&infty.} divergent

2) _ _ _ _ _ _ ~a_~n _ = _ fract{1,~n^2} , _ _ _ _ _ _ compare with _ ~b_~n = fract{1,~n ( ~n + 1 )}

fract{~a_~n,~b_~n} _ = _ fract{~n ( ~n + 1 ),~n^2} _ _ _ _ --> _ _ _ _ 1

Now _ _ fract{1,~n ( ~n + 1 )} _ = _ fract{1,~n} - fract{1, ~n + 1 }

i.e. _ _ sum{~b_~n,1,&infty.} _ = _ 1 - fract{1,2} + fract{1,2} - fract{1,3} + fract{1,3} - ...

so the partial sum _ _ ~s_~k _ = _ 1 - fract{1, ~k + 1 } _ _ -> _ _ 1 _ as _ ~n _ -> _ &infty.

so _ sum{fract{1,~n^2},1,&infty.} _ is also convergent.

Exponential Series

Let

~b_~n _ = _ script{rndb{1 + fract{1,~n}},,,~n,} _ = _ sum{comb{~n,~i} (1/~n)^~i,~i = 0,~n}

or, more explicitly

_ = _ sum{fract{~n#!,(~n - ~i)#! ~i #!} fract{1,~n^~i},~i = 0,~n} _ = _ sum{fract{~n#!,(~n - ~i)#! ~n^~i} fract{1,~i #!},~i = 0,~n}

Consider the ~i^{th} term of the sum:

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

_ = _ rndb{fract{~n,~n}fract{~n - 1,~n}fract{~n - 2,~n} ... fract{~n - ~i + 1,~n}} fract{1,~i #!}

_ = _ rndb{1 # rndb{1 - fract{1,~n}} # ... # rndb{1 - fract{~i - 1,~n}}} fract{1,~i #!} _ _ _ _ < _ _ _ _ fract{1,~i #!}

Clearly _ ~a_~i^{(~n + 1)} _ > _ ~a_~i^{(~n)} , _ ~i = 1 ... ~n , _ _ [ since ( 1 - ( ~k / (~n + 1) )) _ > _ ( 1 - ( ~k / ~n )) _ ~k &in. &naturals. ] .
also ~b_{~n + 1} has one more positive term than ~b_~n _ ( i.e. _ ~a_{~n + 1}^{(~n + 1)} ), _ so ~b_{~n + 1} _ > _ ~b_~n , _ i.e. \{~b_~n\} increasing.
But _ ~a_~i^{( ~n )} _ < _ 1 / ~i #! , _ so

~b_~n _ _ < _ _ sum{fract{1,~i #!},~i = 0,~n} _ _ < _ _ sum{fract{1,~i #!},~i = 0,&infty.} _ _ =#: _ _ ~e

so ~b_~n increasing sequence bounded by ~e. _ ~b_~n converges to some number less than or equal to ~e

~e _ _ _ _ >= _ _ _ _ lim{script{rndb{1 + fract{1,~n}},,,~n,},~n -> &infty.}

Note that _ ~b_~n _ is ~{not} a partial sum in the defined sense, as _ ~b_~n _ is not _ ~b_~{n - 1} _ plus just one extra term!

2 < ~e < 3 , _ since

~e _ = _ sum{fract{1,~i #!},~i = 0,&infty.} _ = _ 1 + fract{1,1 #!} + sum{fract{1,~i #!},~i = 2,&infty.} _ _ > _ _ 2

~e _ = _ sum{fract{1,~i #!},~i = 0,&infty.} _ = _ 1 + sum{fract{1,~i #!},~i = 1,&infty.} _ _ < _ _ 1 + sum{fract{1,2^{~i - 1}},~i = 1,&infty.} _ = _ 3

Since ~i #! _ >= _ 2^{~i - 1} , _ by induction, and the last sum is an example of the geometric series

Now _ _ ~b_~n _ = _ sum{~a_~i^{(~n )},~i = 0,~n} , _ _ and for any ~{fixed} _ ~i , _ _ ~a_~i^{(~n )} -> 1 / ~i #! _ as ~n -> &infty. . _ _ But the limit for _ ~b_~n _ involves terms with infinitely large ~i , _ so taking the limit of the sum as the sum of limits of the terms is a bit awkward in this case.

Instead we will later use integral theory to show that ~e is the limit of ( 1 + ( 1 / ~n ) )^~n .