Mixing
the series $sum_{n=1}^{infty}f_n(x)$ converges uniformly Choose the correct option
let $f :mathbb{R} rightarrow mathbb{R}$ be non zero function such that $|f(x)| le frac{1}{1+2x^2}$ for all $x in mathbb{R}$ .Define real value function $f_n$ on $mathbb{R}$ for all $n in mathbb{N}$ by $f_n(x) = f(x +n)$.
Then ,the series $sum_{n=1}^{infty}f_n(x)$ converges uniformly
Choose the correct option
$a)$ on $[0,1]$
$b)$ on $[-1,0]$
$c)$ on $[-1,1]$
$d)$ None of these
I thinks option $a) ,b)$ and $c)$ will correct because derivative of $f$ is bounded
Is its True??
real-analysis
asked Nov 21 '18 at 0:51
Messi fifa
51611
add a comment |
let $f :mathbb{R} rightarrow mathbb{R}$ be non zero function such that $|f(x)| le frac{1}{1+2x^2}$ for all $x in mathbb{R}$ .Define real value function $f_n$ on $mathbb{R}$ for all $n in mathbb{N}$ by $f_n(x) = f(x +n)$.
Then ,the series $sum_{n=1}^{infty}f_n(x)$ converges uniformly
Choose the correct option
$a)$ on $[0,1]$
$b)$ on $[-1,0]$
$c)$ on $[-1,1]$
$d)$ None of these
I thinks option $a) ,b)$ and $c)$ will correct because derivative of $f$ is bounded
Is its True??
real-analysis
asked Nov 21 '18 at 0:51
Messi fifa
51611
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04
add a comment |
let $f :mathbb{R} rightarrow mathbb{R}$ be non zero function such that $|f(x)| le frac{1}{1+2x^2}$ for all $x in mathbb{R}$ .Define real value function $f_n$ on $mathbb{R}$ for all $n in mathbb{N}$ by $f_n(x) = f(x +n)$.
Then ,the series $sum_{n=1}^{infty}f_n(x)$ converges uniformly
Choose the correct option
$a)$ on $[0,1]$
$b)$ on $[-1,0]$
$c)$ on $[-1,1]$
$d)$ None of these
I thinks option $a) ,b)$ and $c)$ will correct because derivative of $f$ is bounded
Is its True??
real-analysis
asked Nov 21 '18 at 0:51
Messi fifa
51611
let $f :mathbb{R} rightarrow mathbb{R}$ be non zero function such that $|f(x)| le frac{1}{1+2x^2}$ for all $x in mathbb{R}$ .Define real value function $f_n$ on $mathbb{R}$ for all $n in mathbb{N}$ by $f_n(x) = f(x +n)$.
Then ,the series $sum_{n=1}^{infty}f_n(x)$ converges uniformly
Choose the correct option
$a)$ on $[0,1]$
$b)$ on $[-1,0]$
$c)$ on $[-1,1]$
$d)$ None of these
I thinks option $a) ,b)$ and $c)$ will correct because derivative of $f$ is bounded
Is its True??
real-analysis
real-analysis
asked Nov 21 '18 at 0:51
Messi fifa
51611
asked Nov 21 '18 at 0:51
Messi fifa
51611
asked Nov 21 '18 at 0:51
Messi fifa
51611
asked Nov 21 '18 at 0:51
Messi fifa
51611
asked Nov 21 '18 at 0:51
Messi fifa
51611
51611
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04
add a comment |
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04
add a comment |
1 Answer
1
active
oldest
votes
HINT:
We are not given that $f$ is differentiable.
So, note that for $xin [-1,1]$
$$|f_n(x)|=|f(x+n)|le frac{1}{1+2(x+n)^2}le frac1{1+2(n-1)^2}$$
Appeal to the Weierstrass M-Test. Can you finish?
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
add a 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%2f3007112%2fthe-series-sum-n-1-inftyf-nx-converges-uniformly-choose-the-correct-op%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
HINT:
We are not given that $f$ is differentiable.
So, note that for $xin [-1,1]$
$$|f_n(x)|=|f(x+n)|le frac{1}{1+2(x+n)^2}le frac1{1+2(n-1)^2}$$
Appeal to the Weierstrass M-Test. Can you finish?
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
add a comment |
HINT:
We are not given that $f$ is differentiable.
So, note that for $xin [-1,1]$
$$|f_n(x)|=|f(x+n)|le frac{1}{1+2(x+n)^2}le frac1{1+2(n-1)^2}$$
Appeal to the Weierstrass M-Test. Can you finish?
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
add a comment |
HINT:
We are not given that $f$ is differentiable.
So, note that for $xin [-1,1]$
$$|f_n(x)|=|f(x+n)|le frac{1}{1+2(x+n)^2}le frac1{1+2(n-1)^2}$$
Appeal to the Weierstrass M-Test. Can you finish?
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
HINT:
We are not given that $f$ is differentiable.
So, note that for $xin [-1,1]$
$$|f_n(x)|=|f(x+n)|le frac{1}{1+2(x+n)^2}le frac1{1+2(n-1)^2}$$
Appeal to the Weierstrass M-Test. Can you finish?
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
answered Nov 21 '18 at 2:15
Mark Viola
130k1274170
130k1274170
add a comment |
add a 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.
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.
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%2f3007112%2fthe-series-sum-n-1-inftyf-nx-converges-uniformly-choose-the-correct-op%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
See $displaystyle f_n<dfrac{1}{1+2x^2} :forall:n$. So what can you say about the sum?
– Yadati Kiran
Nov 21 '18 at 1:04