Right-sided derivative of $f(x)=fracsin(sqrtx)sqrtx$Power series, derivatives, integrals, and different intervals of convergenceRadius of convergence two power series (by using Cauchy test).Convergence radius of power series is infiniteRight derivative of a power seriesRadius of convergence of $sum_n=1^infty sin(sqrtn+1 - sqrtn)(x-2)^n$Showing the radius of convergence for a power series is equal to the radius of convergence for its derivativeClosed form for Binomial Power SeriesDoes one-sided derivative of real power series at edge of domain of convergenceExplanation on differentiating power seriesPower Series, Taylor/Maclaurin and n-th derivative
How to Prove P(a) → ∀x(P(x) ∨ ¬(x = a)) using Natural Deduction
Blending or harmonizing
Mathematica command that allows it to read my intentions
How many wives did king shaul have
How could indestructible materials be used in power generation?
OP Amp not amplifying audio signal
What are the G forces leaving Earth orbit?
How do I exit BASH while loop using modulus operator?
Knowledge-based authentication using Domain-driven Design in C#
Venezuelan girlfriend wants to travel the USA to be with me. What is the process?
Can a virus destroy the BIOS of a modern computer?
How do conventional missiles fly?
How to compactly explain secondary and tertiary characters without resorting to stereotypes?
Using "tail" to follow a file without displaying the most recent lines
Should I tell management that I intend to leave due to bad software development practices?
Why do I get negative height?
What is a Samsaran Word™?
Rotate ASCII Art by 45 Degrees
Car headlights in a world without electricity
Do creatures with a listed speed of "0 ft., fly 30 ft. (hover)" ever touch the ground?
Why are UK visa biometrics appointments suspended at USCIS Application Support Centers?
How to prevent "they're falling in love" trope
How can I deal with my CEO asking me to hire someone with a higher salary than me, a co-founder?
Placement of More Information/Help Icon button for Radio Buttons
Right-sided derivative of $f(x)=fracsin(sqrtx)sqrtx$
Power series, derivatives, integrals, and different intervals of convergenceRadius of convergence two power series (by using Cauchy test).Convergence radius of power series is infiniteRight derivative of a power seriesRadius of convergence of $sum_n=1^infty sin(sqrtn+1 - sqrtn)(x-2)^n$Showing the radius of convergence for a power series is equal to the radius of convergence for its derivativeClosed form for Binomial Power SeriesDoes one-sided derivative of real power series at edge of domain of convergenceExplanation on differentiating power seriesPower Series, Taylor/Maclaurin and n-th derivative
$begingroup$
Let define $f:mathbbRrightarrowmathbbR$ such as
$f(x)=fracsin(sqrtx)sqrtx$ for $x>0$ and $f(x)=1$ for $x=0$
Prove that $f$ has right-sided all-order derivatives in $0$.
My approach:
I find out that $f$ can be written as the power series
$$
f(x)=sum_n=0^inftyfrac(-1)^k(2k+1)!x^k
$$
which has an infinite radius of convergence so the $n$-th derivative of $f$ in zero would be
$$
f^(n)(0)=frac(-1)^n n!(2n+1)!
$$
but I do not know how to connect these facts with the right-sided derivative of this function
real-analysis derivatives trigonometry power-series
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
$endgroup$
add a comment |
$begingroup$
Let define $f:mathbbRrightarrowmathbbR$ such as
$f(x)=fracsin(sqrtx)sqrtx$ for $x>0$ and $f(x)=1$ for $x=0$
Prove that $f$ has right-sided all-order derivatives in $0$.
My approach:
I find out that $f$ can be written as the power series
$$
f(x)=sum_n=0^inftyfrac(-1)^k(2k+1)!x^k
$$
which has an infinite radius of convergence so the $n$-th derivative of $f$ in zero would be
$$
f^(n)(0)=frac(-1)^n n!(2n+1)!
$$
but I do not know how to connect these facts with the right-sided derivative of this function
real-analysis derivatives trigonometry power-series
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
$endgroup$
add a comment |
$begingroup$
Let define $f:mathbbRrightarrowmathbbR$ such as
$f(x)=fracsin(sqrtx)sqrtx$ for $x>0$ and $f(x)=1$ for $x=0$
Prove that $f$ has right-sided all-order derivatives in $0$.
My approach:
I find out that $f$ can be written as the power series
$$
f(x)=sum_n=0^inftyfrac(-1)^k(2k+1)!x^k
$$
which has an infinite radius of convergence so the $n$-th derivative of $f$ in zero would be
$$
f^(n)(0)=frac(-1)^n n!(2n+1)!
$$
but I do not know how to connect these facts with the right-sided derivative of this function
real-analysis derivatives trigonometry power-series
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
$endgroup$
Let define $f:mathbbRrightarrowmathbbR$ such as
$f(x)=fracsin(sqrtx)sqrtx$ for $x>0$ and $f(x)=1$ for $x=0$
Prove that $f$ has right-sided all-order derivatives in $0$.
My approach:
I find out that $f$ can be written as the power series
$$
f(x)=sum_n=0^inftyfrac(-1)^k(2k+1)!x^k
$$
which has an infinite radius of convergence so the $n$-th derivative of $f$ in zero would be
$$
f^(n)(0)=frac(-1)^n n!(2n+1)!
$$
but I do not know how to connect these facts with the right-sided derivative of this function
real-analysis derivatives trigonometry power-series
real-analysis derivatives trigonometry power-series
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
edited Mar 28 at 16:23
José Carlos Santos
172k23132240
172k23132240
asked Mar 28 at 16:03
avan1235avan1235
3578
asked Mar 28 at 16:03
avan1235avan1235
3578
asked Mar 28 at 16:03
avan1235avan1235
3578
3578
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since$$sin(x)=x-fracx^33!+fracx^55!-fracx^77!+cdots,$$you have$$fracsinleft(sqrt xright)sqrt x=1-frac x3!+fracx^25!-fracx^37!+cdots$$when $xgeqslant0$, from which it follows that, indeed, $f$ has right-sided derivatives of all orders at $0$.
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
|
show 1 more comment
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166094%2fright-sided-derivative-of-fx-frac-sin-sqrtx-sqrtx%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
$begingroup$
Since$$sin(x)=x-fracx^33!+fracx^55!-fracx^77!+cdots,$$you have$$fracsinleft(sqrt xright)sqrt x=1-frac x3!+fracx^25!-fracx^37!+cdots$$when $xgeqslant0$, from which it follows that, indeed, $f$ has right-sided derivatives of all orders at $0$.
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
|
show 1 more comment
$begingroup$
Since$$sin(x)=x-fracx^33!+fracx^55!-fracx^77!+cdots,$$you have$$fracsinleft(sqrt xright)sqrt x=1-frac x3!+fracx^25!-fracx^37!+cdots$$when $xgeqslant0$, from which it follows that, indeed, $f$ has right-sided derivatives of all orders at $0$.
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
|
show 1 more comment
$begingroup$
Since$$sin(x)=x-fracx^33!+fracx^55!-fracx^77!+cdots,$$you have$$fracsinleft(sqrt xright)sqrt x=1-frac x3!+fracx^25!-fracx^37!+cdots$$when $xgeqslant0$, from which it follows that, indeed, $f$ has right-sided derivatives of all orders at $0$.
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
$endgroup$
Since$$sin(x)=x-fracx^33!+fracx^55!-fracx^77!+cdots,$$you have$$fracsinleft(sqrt xright)sqrt x=1-frac x3!+fracx^25!-fracx^37!+cdots$$when $xgeqslant0$, from which it follows that, indeed, $f$ has right-sided derivatives of all orders at $0$.
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
answered Mar 28 at 16:07
José Carlos SantosJosé Carlos Santos
172k23132240
172k23132240
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
|
show 1 more comment
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
what theoem stands for your statement that $f$ has right-sided derivatives of all orders?
$endgroup$
– avan1235
Mar 28 at 16:33
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
The theorem that says that if a power series $sum_n=0^infty a_n(x-a)^n$ centered at $a$ has non-zero radius of convergence, then $(forall ninmathbbZ^+):fracf^(n)(a)n!=a_n$.
$endgroup$
– José Carlos Santos
Mar 28 at 16:36
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
But I know only this theorem for the normal limits not sided one. How to connect this theorem with this special case?
$endgroup$
– avan1235
Mar 28 at 17:20
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
If the derivatives exist for both sides, then one-sided derivatives exit too.
$endgroup$
– José Carlos Santos
Mar 28 at 17:30
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
$begingroup$
But what theorem (name) stands for that we can connect the given series (that converges for all values, also these smaller than zero) with the initial function (which can be defined only for positive numbers)
$endgroup$
– avan1235
Mar 28 at 19:34
|
show 1 more comment
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166094%2fright-sided-derivative-of-fx-frac-sin-sqrtx-sqrtx%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
