Fdtxjr

Multi tool use

How do I locate a classical quotation?

In the late 1940’s to early 1950’s what technology was available that could melt a LOT of ice?

Best approach to update all entries in a list that is paginated?

Aliens englobed the Solar System: will we notice?

Rejected in 4th interview round citing insufficient years of experience

Could you please stop shuffling the deck and play already?

Latest web browser compatible with Windows 98

Are the terms "stab" and "staccato" synonyms?

Force user to remove USB token

Subset counting for even numbers

Why doesn't this Google Translate ad use the word "Translation" instead of "Translate"?

Do f-stop and exposure time perfectly cancel?

Why would one plane in this picture not have gear down yet?

If the Captain's screens are out, does he switch seats with the co-pilot?

Why does Deadpool say "You're welcome, Canada," after shooting Ryan Reynolds in the end credits?

What are some noteworthy "mic-drop" moments in math?

Why would a jet engine that runs at temps excess of 2000°C burn when it crashes?

Low budget alien movie about the Earth being cooked

How does airport security verify that you can carry a battery bank over 100 Wh?

Does splitting a potentially monolithic application into several smaller ones help prevent bugs?

Space in array system equations

Unreachable code, but reachable with exception

Is "history" a male-biased word ("his+story")?

Single word request: Harming the benefactor

Solution verification: finding a Maclaurin series for $f$, interval of convergence, and $f^(10)(0)$

Question about Maclaurin Series for $cos x$Taylor and Maclaurin Series for $f(x)=e^x$Maclaurin series for $e^x +2e^-x$Maclaurin series of $sin(2pi x)$Finding interval of convergence for seriesMaclaurin Series expansion intervalMaclaurin Series for a natural logarithmMaclaurin series - Approximation and interval of convergenceMaclaurin series for lnMaclaurin Series from sin(x) to cos(x) using derivative

2

$begingroup$

I have to find Maclaurin series for function $f(x)$ = $2x^2over16+x^4$, it's interval of convergence and $f^(10)(0)$. I managed to calculate Maclaurin series and $10^th$ derivative, but I'm not sure if it's done in proper way and if solution is correct.

---

On determining the series,

$$beginalign
f(x) &= frac2x^216+x^4 \
&= frac2x^216 cdot frac11-frac-x^416 \
&= frac2x^216 cdot sum_i=0^infty left(-fracx^416 right)^n \
&= frac2x^216 cdot sum_i=0^infty frac(-1)^n cdot x^4n16^n \
&= sum_i=0^infty frac(-1)^n cdot x^4n+22^4n+3
endalign$$

for $vert-x^4over16vert<1 implies xin(-2;2)$

---

On determining $f^(10)(0)$,

$$f^(10)(0)cdot frac x^10 10! = fracx^422^43 implies f^(10)(0) = x^32cdot frac10!2^43$$

---

Looking for feedback and opinion if the way I solved it is correct.

calculus proof-verification taylor-expansion

share|cite|improve this question

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

$endgroup$

add a comment | 

2

$begingroup$

I have to find Maclaurin series for function $f(x)$ = $2x^2over16+x^4$, it's interval of convergence and $f^(10)(0)$. I managed to calculate Maclaurin series and $10^th$ derivative, but I'm not sure if it's done in proper way and if solution is correct.

---

On determining the series,

$$beginalign
f(x) &= frac2x^216+x^4 \
&= frac2x^216 cdot frac11-frac-x^416 \
&= frac2x^216 cdot sum_i=0^infty left(-fracx^416 right)^n \
&= frac2x^216 cdot sum_i=0^infty frac(-1)^n cdot x^4n16^n \
&= sum_i=0^infty frac(-1)^n cdot x^4n+22^4n+3
endalign$$

for $vert-x^4over16vert<1 implies xin(-2;2)$

---

On determining $f^(10)(0)$,

$$f^(10)(0)cdot frac x^10 10! = fracx^422^43 implies f^(10)(0) = x^32cdot frac10!2^43$$

---

Looking for feedback and opinion if the way I solved it is correct.

calculus proof-verification taylor-expansion

share|cite|improve this question

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

$endgroup$

add a comment | 

$begingroup$

I have to find Maclaurin series for function $f(x)$ = $2x^2over16+x^4$, it's interval of convergence and $f^(10)(0)$. I managed to calculate Maclaurin series and $10^th$ derivative, but I'm not sure if it's done in proper way and if solution is correct.

---

On determining the series,

$$beginalign
f(x) &= frac2x^216+x^4 \
&= frac2x^216 cdot frac11-frac-x^416 \
&= frac2x^216 cdot sum_i=0^infty left(-fracx^416 right)^n \
&= frac2x^216 cdot sum_i=0^infty frac(-1)^n cdot x^4n16^n \
&= sum_i=0^infty frac(-1)^n cdot x^4n+22^4n+3
endalign$$

for $vert-x^4over16vert<1 implies xin(-2;2)$

---

On determining $f^(10)(0)$,

$$f^(10)(0)cdot frac x^10 10! = fracx^422^43 implies f^(10)(0) = x^32cdot frac10!2^43$$

---

Looking for feedback and opinion if the way I solved it is correct.

calculus proof-verification taylor-expansion

share|cite|improve this question

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

$endgroup$

share|cite|improve this question

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

share|cite|improve this question

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

edited 2 days ago

Eevee Trainer

7,80621339

edited 2 days ago

Eevee Trainer

7,80621339

asked 2 days ago

MichaelMichael

246

asked 2 days ago

MichaelMichael

246

1 Answer
1

active

oldest

votes

2

$begingroup$

Your computation of the MacLaurin series of $f$ looks right, but remember that $f^(10)(0)$ is a number. You know that$$fracf^(10)(0)10!=frac(-1)^22^11.$$Therefore,$$f^(10)(0)=frac10!2^11.$$

Also, $leftlvertfrac-x^416rightrvert<1$ doesn't just imply that $xin(-2,2)$; it is actually equivalent to it.

share|cite|improve this answer

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What happened to x is substituting it with 2 and -2 (what I got from interval of convergence)?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The series diverges at those two points.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I’m ahead with the program and doing it Maclaurin on my own, so sorry for tedious questions. To get 10th derivative you input -2 or 2? Is that the way to calculate it?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  To get the $10^textth$ derivative, I put $n=2$, so that $x^4n+2$ becomes $x^10$.
  $endgroup$
  – José Carlos Santos
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad I could help.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  

 | 
show 5 more comments

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

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3142211%2fsolution-verification-finding-a-maclaurin-series-for-f-interval-of-convergen%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

Your computation of the MacLaurin series of $f$ looks right, but remember that $f^(10)(0)$ is a number. You know that$$fracf^(10)(0)10!=frac(-1)^22^11.$$Therefore,$$f^(10)(0)=frac10!2^11.$$

Also, $leftlvertfrac-x^416rightrvert<1$ doesn't just imply that $xin(-2,2)$; it is actually equivalent to it.

share|cite|improve this answer

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What happened to x is substituting it with 2 and -2 (what I got from interval of convergence)?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The series diverges at those two points.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I’m ahead with the program and doing it Maclaurin on my own, so sorry for tedious questions. To get 10th derivative you input -2 or 2? Is that the way to calculate it?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  To get the $10^textth$ derivative, I put $n=2$, so that $x^4n+2$ becomes $x^10$.
  $endgroup$
  – José Carlos Santos
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad I could help.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  

 | 
show 5 more comments

2

$begingroup$

Your computation of the MacLaurin series of $f$ looks right, but remember that $f^(10)(0)$ is a number. You know that$$fracf^(10)(0)10!=frac(-1)^22^11.$$Therefore,$$f^(10)(0)=frac10!2^11.$$

Also, $leftlvertfrac-x^416rightrvert<1$ doesn't just imply that $xin(-2,2)$; it is actually equivalent to it.

share|cite|improve this answer

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What happened to x is substituting it with 2 and -2 (what I got from interval of convergence)?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The series diverges at those two points.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I’m ahead with the program and doing it Maclaurin on my own, so sorry for tedious questions. To get 10th derivative you input -2 or 2? Is that the way to calculate it?
  $endgroup$
  – Michael
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  To get the $10^textth$ derivative, I put $n=2$, so that $x^4n+2$ becomes $x^10$.
  $endgroup$
  – José Carlos Santos
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad I could help.
  $endgroup$
  – José Carlos Santos
  2 days ago
  

  
  
  
  

 | 
show 5 more comments

$begingroup$

Your computation of the MacLaurin series of $f$ looks right, but remember that $f^(10)(0)$ is a number. You know that$$fracf^(10)(0)10!=frac(-1)^22^11.$$Therefore,$$f^(10)(0)=frac10!2^11.$$

Also, $leftlvertfrac-x^416rightrvert<1$ doesn't just imply that $xin(-2,2)$; it is actually equivalent to it.

share|cite|improve this answer

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

$endgroup$

share|cite|improve this answer

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235

answered 2 days ago

José Carlos SantosJosé Carlos Santos

167k22132235