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From: Maury Barbato on 14 Apr 2010 01:27
Hello,
let

sum_{n=0}^Infty (a_n)*(x^n)

be a real power series with radius of convergence R > 0.
Set, for every -R < x < R,

f(x) = sum_{n=0}^Infty (a_n)*(x^n).

Then, for every m>=0, let

sum_{n=0}^Infty (b_{n, m})*(x^n)

a real power series with radius of convergence R_m >= R.
Set, for every -R_m < x < R_m,

f_m(x) = sum_{n=0}^Infty (b_{n, m})*(x^n),

ans suppose that, for some 0 < r < R, the series of
functions {f_m(x)} converges pointwise to f(x) in
[-r, r]. Can we conclude that, for every n>= 0,

lim_{m -> Infty} b_{n, m} = a_n ?

What if we suppose uniform convergence in [-r,r]?

Thank you very much for your attention.
My Best Regards,
Maury Barbato
From: José Carlos Santos on 14 Apr 2010 06:11
On 14-04-2010 10:27, Maury Barbato wrote:

> let
>
> sum_{n=0}^Infty (a_n)*(x^n)
>
> be a real power series with radius of convergence R> 0.
> Set, for every -R< x< R,
>
> f(x) = sum_{n=0}^Infty (a_n)*(x^n).
>
> Then, for every m>=0, let
>
> sum_{n=0}^Infty (b_{n, m})*(x^n)
>
> a real power series with radius of convergence R_m>= R.
> Set, for every -R_m< x< R_m,
>
> f_m(x) = sum_{n=0}^Infty (b_{n, m})*(x^n),
>
> ans suppose that, for some 0< r< R, the series of
> functions {f_m(x)} converges pointwise to f(x) in
> [-r, r]. Can we conclude that, for every n>= 0,
>
> lim_{m -> Infty} b_{n, m} = a_n ?

No. Take f(x) = 0 and f_m(x) = sin(n*x)/n. Then (f_m)_m converges
pointwise to _f_, but each b_{1,m} is equal to 1. Therefore, the
sequence (b_{1,m})_m does not converge to 0 (=a_1).

> What if we suppose uniform convergence in [-r,r]?

See my previous example.

However, the answer is affirmative if we assume that the convergence
is uniform on some _disk_ (from the complex plane) centered at 0.

Best regards,

Jose Carlos Santos
From: José Carlos Santos on 14 Apr 2010 06:12
On 14-04-2010 11:11, Jos� Carlos Santos wrote:

>> let
>>
>> sum_{n=0}^Infty (a_n)*(x^n)
>>
>> be a real power series with radius of convergence R> 0.
>> Set, for every -R< x< R,
>>
>> f(x) = sum_{n=0}^Infty (a_n)*(x^n).
>>
>> Then, for every m>=0, let
>>
>> sum_{n=0}^Infty (b_{n, m})*(x^n)
>>
>> a real power series with radius of convergence R_m>= R.
>> Set, for every -R_m< x< R_m,
>>
>> f_m(x) = sum_{n=0}^Infty (b_{n, m})*(x^n),
>>
>> ans suppose that, for some 0< r< R, the series of
>> functions {f_m(x)} converges pointwise to f(x) in
>> [-r, r]. Can we conclude that, for every n>= 0,
>>
>> lim_{m -> Infty} b_{n, m} = a_n ?
>
> No. Take f(x) = 0 and f_m(x) = sin(n*x)/n.

Of course, what I meant was sin(m*x)/m.

Best regards,

Jose Carlos Santos

From: Maury Barbato on 14 Apr 2010 02:44
Jose Carlos wrote:

> On 14-04-2010 10:27, Maury Barbato wrote:
>
> > let
> >
> > sum_{n=0}^Infty (a_n)*(x^n)
> >
> > be a real power series with radius of convergence
> R> 0.
> > Set, for every -R< x< R,
> >
> > f(x) = sum_{n=0}^Infty (a_n)*(x^n).
> >
> > Then, for every m>=0, let
> >
> > sum_{n=0}^Infty (b_{n, m})*(x^n)
> >
> > a real power series with radius of convergence
> R_m>= R.
> > Set, for every -R_m< x< R_m,
> >
> > f_m(x) = sum_{n=0}^Infty (b_{n, m})*(x^n),
> >
> > ans suppose that, for some 0< r< R, the series of
> > functions {f_m(x)} converges pointwise to f(x) in
> > [-r, r]. Can we conclude that, for every n>= 0,
> >
> > lim_{m -> Infty} b_{n, m} = a_n ?
>
> No. Take f(x) = 0 and f_m(x) = sin(n*x)/n. Then
> (f_m)_m converges
> pointwise to _f_, but each b_{1,m} is equal to 1.
> Therefore, the
> sequence (b_{1,m})_m does not converge to 0 (=a_1).
>
> > What if we suppose uniform convergence in [-r,r]?
>
> See my previous example.
>
> However, the answer is affirmative if we assume that
> the convergence
> is uniform on some _disk_ (from the complex plane)
> centered at 0.
>
> Best regards,
>
> Jose Carlos Santos

Thank you very very ... much, Jose.
Your examples are enlithning!

Friendly Regards,
Maury Barbato

Amo
todas
las cosas,
un porque sean
ardientes
o fragantes,
sino porque
no sé,
porque
este océano es el tuyo,
es el mío:
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