show that they have the same formal power series coefficients












1














$ underline{text{This is about formal power series:}}$



If two power series (in one or two variables) converge on a disc and agree on infinitely many values within that disc, then show that they have the same formal power series coefficients.



Answer:



Let $f(x)=sum_{n geq 0} a_n x^n$ and $g(x)=sum_{n geq 0} b_n x^n$ be two power series converges on the disc $D(r^{-})={x in mathbb{C}: |x|<r }$.



Their corresponding formal power series are respectively $ F(x)=sum_{n geq 0} a_n X^n$ and $G(x)=sum_{n geq 0} b_n X^n $, say.



Given that the power series agree on $D(r^{-})$, so we have



$sum_{n geq 0} a_n x^n=sum_{n geq 0} b_n x^n$ for infinitely many $x in D(r^{-})$,



Does this imply $ sum_{n geq 0} a_n X^n=sum_{n geq 0} a_n X^n$?



If so then, let



$h(X)=sum_{n geq 0} a_n X^n-sum_{n geq 0} b_n X^n=0$,



$Rightarrow h(X)=0$



$Rightarrow sum_{n geq 0} (a_n-b_n) X^n=0$,



$Rightarrow a_n=b_n, forall n$



I think some mistake happened .



Please help me






power-series formal-power-series







share|cite|improve this question
























  • Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
    – Somos
    5 mins ago


















1














$ underline{text{This is about formal power series:}}$



If two power series (in one or two variables) converge on a disc and agree on infinitely many values within that disc, then show that they have the same formal power series coefficients.



Answer:



Let $f(x)=sum_{n geq 0} a_n x^n$ and $g(x)=sum_{n geq 0} b_n x^n$ be two power series converges on the disc $D(r^{-})={x in mathbb{C}: |x|<r }$.



Their corresponding formal power series are respectively $ F(x)=sum_{n geq 0} a_n X^n$ and $G(x)=sum_{n geq 0} b_n X^n $, say.



Given that the power series agree on $D(r^{-})$, so we have



$sum_{n geq 0} a_n x^n=sum_{n geq 0} b_n x^n$ for infinitely many $x in D(r^{-})$,



Does this imply $ sum_{n geq 0} a_n X^n=sum_{n geq 0} a_n X^n$?



If so then, let



$h(X)=sum_{n geq 0} a_n X^n-sum_{n geq 0} b_n X^n=0$,



$Rightarrow h(X)=0$



$Rightarrow sum_{n geq 0} (a_n-b_n) X^n=0$,



$Rightarrow a_n=b_n, forall n$



I think some mistake happened .



Please help me






power-series formal-power-series







share|cite|improve this question
























  • Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
    – Somos
    5 mins ago
















1












1








1


2





$ underline{text{This is about formal power series:}}$



If two power series (in one or two variables) converge on a disc and agree on infinitely many values within that disc, then show that they have the same formal power series coefficients.



Answer:



Let $f(x)=sum_{n geq 0} a_n x^n$ and $g(x)=sum_{n geq 0} b_n x^n$ be two power series converges on the disc $D(r^{-})={x in mathbb{C}: |x|<r }$.



Their corresponding formal power series are respectively $ F(x)=sum_{n geq 0} a_n X^n$ and $G(x)=sum_{n geq 0} b_n X^n $, say.



Given that the power series agree on $D(r^{-})$, so we have



$sum_{n geq 0} a_n x^n=sum_{n geq 0} b_n x^n$ for infinitely many $x in D(r^{-})$,



Does this imply $ sum_{n geq 0} a_n X^n=sum_{n geq 0} a_n X^n$?



If so then, let



$h(X)=sum_{n geq 0} a_n X^n-sum_{n geq 0} b_n X^n=0$,



$Rightarrow h(X)=0$



$Rightarrow sum_{n geq 0} (a_n-b_n) X^n=0$,



$Rightarrow a_n=b_n, forall n$



I think some mistake happened .



Please help me






power-series formal-power-series







share|cite|improve this question















$ underline{text{This is about formal power series:}}$



If two power series (in one or two variables) converge on a disc and agree on infinitely many values within that disc, then show that they have the same formal power series coefficients.



Answer:



Let $f(x)=sum_{n geq 0} a_n x^n$ and $g(x)=sum_{n geq 0} b_n x^n$ be two power series converges on the disc $D(r^{-})={x in mathbb{C}: |x|<r }$.



Their corresponding formal power series are respectively $ F(x)=sum_{n geq 0} a_n X^n$ and $G(x)=sum_{n geq 0} b_n X^n $, say.



Given that the power series agree on $D(r^{-})$, so we have



$sum_{n geq 0} a_n x^n=sum_{n geq 0} b_n x^n$ for infinitely many $x in D(r^{-})$,



Does this imply $ sum_{n geq 0} a_n X^n=sum_{n geq 0} a_n X^n$?



If so then, let



$h(X)=sum_{n geq 0} a_n X^n-sum_{n geq 0} b_n X^n=0$,



$Rightarrow h(X)=0$



$Rightarrow sum_{n geq 0} (a_n-b_n) X^n=0$,



$Rightarrow a_n=b_n, forall n$



I think some mistake happened .



Please help me






power-series formal-power-series




power-series formal-power-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago

























asked 2 hours ago









M. A. SARKAR

2,1771619




2,1771619












  • Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
    – Somos
    5 mins ago




















  • Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
    – Somos
    5 mins ago


















Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
– Somos
5 mins ago






Given $f(x)=sum_{n geq 0} a_n x^n$, then what do you mean by its corresponding formal power series is $F(x)=sum_{n geq 0} a_n X^n$? Now, the left side is a function of $x$ and the right of $X$ If you really meant $x$ on the right side, then $f(x)=F(x)$ by definition.
– Somos
5 mins ago












1 Answer
1






active

oldest

votes


















4














Not true. Let $f(z)=sin (frac 1 {z-1})$. This function has a power series expansion $sum a_n z^{n}$ in ${z:|z| <1}$. The sum of its series is $0 (equiv sum b_nz^{n}$ where $b_n=0$ for all $n$) at the points $z=1- frac 1 {npi}, n=1,2,cdots$. But its coefficients are not all $0$.






share|cite|improve this answer





















  • Then what would be the answer regarding formal power series?
    – M. A. SARKAR
    1 hour ago










  • remember there is the concept of formal power series. please read the question again
    – M. A. SARKAR
    1 hour ago










  • You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
    – Kavi Rama Murthy
    1 hour ago








  • 1




    @KaviRamaMurthy, your answer did not complete proof of the question
    – arifamath
    1 hour ago






  • 1




    Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
    – jmerry
    1 hour ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058322%2fshow-that-they-have-the-same-formal-power-series-coefficients%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4














Not true. Let $f(z)=sin (frac 1 {z-1})$. This function has a power series expansion $sum a_n z^{n}$ in ${z:|z| <1}$. The sum of its series is $0 (equiv sum b_nz^{n}$ where $b_n=0$ for all $n$) at the points $z=1- frac 1 {npi}, n=1,2,cdots$. But its coefficients are not all $0$.






share|cite|improve this answer





















  • Then what would be the answer regarding formal power series?
    – M. A. SARKAR
    1 hour ago










  • remember there is the concept of formal power series. please read the question again
    – M. A. SARKAR
    1 hour ago










  • You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
    – Kavi Rama Murthy
    1 hour ago








  • 1




    @KaviRamaMurthy, your answer did not complete proof of the question
    – arifamath
    1 hour ago






  • 1




    Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
    – jmerry
    1 hour ago
















4














Not true. Let $f(z)=sin (frac 1 {z-1})$. This function has a power series expansion $sum a_n z^{n}$ in ${z:|z| <1}$. The sum of its series is $0 (equiv sum b_nz^{n}$ where $b_n=0$ for all $n$) at the points $z=1- frac 1 {npi}, n=1,2,cdots$. But its coefficients are not all $0$.






share|cite|improve this answer





















  • Then what would be the answer regarding formal power series?
    – M. A. SARKAR
    1 hour ago










  • remember there is the concept of formal power series. please read the question again
    – M. A. SARKAR
    1 hour ago










  • You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
    – Kavi Rama Murthy
    1 hour ago








  • 1




    @KaviRamaMurthy, your answer did not complete proof of the question
    – arifamath
    1 hour ago






  • 1




    Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
    – jmerry
    1 hour ago














4












4








4






Not true. Let $f(z)=sin (frac 1 {z-1})$. This function has a power series expansion $sum a_n z^{n}$ in ${z:|z| <1}$. The sum of its series is $0 (equiv sum b_nz^{n}$ where $b_n=0$ for all $n$) at the points $z=1- frac 1 {npi}, n=1,2,cdots$. But its coefficients are not all $0$.






share|cite|improve this answer












Not true. Let $f(z)=sin (frac 1 {z-1})$. This function has a power series expansion $sum a_n z^{n}$ in ${z:|z| <1}$. The sum of its series is $0 (equiv sum b_nz^{n}$ where $b_n=0$ for all $n$) at the points $z=1- frac 1 {npi}, n=1,2,cdots$. But its coefficients are not all $0$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 hours ago









Kavi Rama Murthy

50.3k31854




50.3k31854












  • Then what would be the answer regarding formal power series?
    – M. A. SARKAR
    1 hour ago










  • remember there is the concept of formal power series. please read the question again
    – M. A. SARKAR
    1 hour ago










  • You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
    – Kavi Rama Murthy
    1 hour ago








  • 1




    @KaviRamaMurthy, your answer did not complete proof of the question
    – arifamath
    1 hour ago






  • 1




    Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
    – jmerry
    1 hour ago


















  • Then what would be the answer regarding formal power series?
    – M. A. SARKAR
    1 hour ago










  • remember there is the concept of formal power series. please read the question again
    – M. A. SARKAR
    1 hour ago










  • You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
    – Kavi Rama Murthy
    1 hour ago








  • 1




    @KaviRamaMurthy, your answer did not complete proof of the question
    – arifamath
    1 hour ago






  • 1




    Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
    – jmerry
    1 hour ago
















Then what would be the answer regarding formal power series?
– M. A. SARKAR
1 hour ago




Then what would be the answer regarding formal power series?
– M. A. SARKAR
1 hour ago












remember there is the concept of formal power series. please read the question again
– M. A. SARKAR
1 hour ago




remember there is the concept of formal power series. please read the question again
– M. A. SARKAR
1 hour ago












You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
– Kavi Rama Murthy
1 hour ago






You talk about formal power series when you don't know that the series converges. Here it is given the the two series converge on a disk.
– Kavi Rama Murthy
1 hour ago






1




1




@KaviRamaMurthy, your answer did not complete proof of the question
– arifamath
1 hour ago




@KaviRamaMurthy, your answer did not complete proof of the question
– arifamath
1 hour ago




1




1




Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
– jmerry
1 hour ago




Does it really have the formal power series you think it does? The series $(z-1)^{-1}-frac1{3!}(z-1)^{-3}+frac1{5!}(z-1)^5-cdots$ doesn't converge in the formal power series sense - there are infinitely many terms with nonzero constant term.
– jmerry
1 hour ago


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058322%2fshow-that-they-have-the-same-formal-power-series-coefficients%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Eastern Orthodox Church

Image
Not to be confused with Oriental Orthodoxy, Eastern Christianity, or Eastern Catholic Churches. "Eastern Orthodoxy", "Orthodox Church", "Orthodox Catholic Church", and "Orthodox Christian Church" redirect here. For other uses, see Orthodox (disambiguation). This article may be too long to read and navigate comfortably . The readable prose size is 132 kilobytes. Please consider splitting content into sub-articles, condensing it, or adding or removing subheadings. ( January 2018 ) Part of a series on the Eastern Orthodox Church Mosaic of Christ Pantocrator, Hagia Sophia Overview Structure Theology (History of theology) Liturgy Church history Holy Mysteries View of salvation View of Mary View of icons Background Crucifixion / Resurrection / Ascension of Jesus Christianity Christian Church Apostolic succession Four Marks of the Church Orthodoxy Organization Autocephaly Patriarchate ...
Read more

Zagreb

Image
This article is about the Croatian capital city. For other uses, see Zagreb (disambiguation). City in City of Zagreb, Croatia Zagreb City Grad Zagreb City of Zagreb Clockwise, from top: St. Mark's Square, Croatian State Archives, Zagreb Tram, Art Pavilion, Cibona & HOTO towers and Croatian National Theatre. Flag Coat of arms Zagreb Location of Zagreb in Croatia Show map of Croatia Zagreb Zagreb (Europe) Show map of Europe Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Country   Croatia County City of Zagreb RC diocese 1094 Free royal city 1242 Unified 1850 Subdivisions 17 city districts 218 local committees (70 settlements) Government  • Type Mayor-Council  • Mayor Milan Bandić (BM 365)  • City Assembly 51 members BM 365, ZL, NS-R (14) SDP, GLAS, HNS, HSS, NH-PS (13) ...
Read more

Understanding the information contained in the Deep Space Network XML data?

Image
up vote 1 down vote favorite NASA's Deep Space Network activity can be viewed conveniently at https://eyes.nasa.gov/dsn/dsn.html but if you would like data in a more raw XML form, it can be accessed as described in this answer Below is a small Python script that reads the XML into a list of dictionaries, and saves it to disk as a JSON. The dictionaries are organized at the higher level by dish, then by spacecraft per dish. Question: What are the most-likely meanings of all the data field contained in the Deep Space Network XML data? This is the example url from the other answer, used in the script below: https://eyes.nasa.gov/dsn/data/1365107113.xml def dictify(r,root=True): # https://stackoverflow.com/questions/2148119/how-to-convert-an-xml-string-to-a-dictionary-in-python/30923963#30923963 if root: ret...
Read more