limit of an infinite sum $y_n$
up vote
-1
down vote
favorite
Let $displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}$. Find $displaystylelim_{nrightarrowinfty}y_n$
I see that the terms in $y_n$ are lie between $displaystylefrac{1}{n}$ and $displaystylefrac{1}{n+1}$.
I am unable to form the telescoping series to get the limit.
P.S. Sorry if this is too basic.
real-analysis sequences-and-series
add a comment |
up vote
-1
down vote
favorite
Let $displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}$. Find $displaystylelim_{nrightarrowinfty}y_n$
I see that the terms in $y_n$ are lie between $displaystylefrac{1}{n}$ and $displaystylefrac{1}{n+1}$.
I am unable to form the telescoping series to get the limit.
P.S. Sorry if this is too basic.
real-analysis sequences-and-series
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}$. Find $displaystylelim_{nrightarrowinfty}y_n$
I see that the terms in $y_n$ are lie between $displaystylefrac{1}{n}$ and $displaystylefrac{1}{n+1}$.
I am unable to form the telescoping series to get the limit.
P.S. Sorry if this is too basic.
real-analysis sequences-and-series
Let $displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}$. Find $displaystylelim_{nrightarrowinfty}y_n$
I see that the terms in $y_n$ are lie between $displaystylefrac{1}{n}$ and $displaystylefrac{1}{n+1}$.
I am unable to form the telescoping series to get the limit.
P.S. Sorry if this is too basic.
real-analysis sequences-and-series
real-analysis sequences-and-series
edited Nov 23 at 18:59
asked Nov 5 at 6:52
Yadati Kiran
1,509518
1,509518
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
4
down vote
accepted
Hint:
You may squeeze $y_n$ as follows:
- For $k=1, ldots , n$ you have
$$frac{n^2}{n^3+2n} leq frac{n^2}{n^3+n+k} leq frac{n^2}{n^3+n}$$
It follows
$$ncdot frac{n^2}{n^3+2n} leq y_n leq n cdot frac{n^2}{n^3+n}$$
Now, take the limits on both sides.
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
add a comment |
up vote
1
down vote
Just added for your curiosity.
If you know harmonic numbers,
$$displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}=n^2 left(H_{n^3+2 n}-H_{n^3+n}right)$$ Now, using, for large $p$
$$H_p=gamma +log left({p}right)+frac{1}{2 p}-frac{1}{12
p^2}+Oleft(frac{1}{p^3}right)$$ and continuing with long division or better with Taylor series for large $n$, you should get
$$y_n=1-frac{3}{2 n^2}+Oleft(frac{1}{n^3}right)$$ For example
$$y_{10}=frac{6646021071305823047954845}{6748980404790313121758326}approx 0.9847$$ while the above approximation would give $frac{197}{200}=0.9850$.
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%2f2985377%2flimit-of-an-infinite-sum-y-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Hint:
You may squeeze $y_n$ as follows:
- For $k=1, ldots , n$ you have
$$frac{n^2}{n^3+2n} leq frac{n^2}{n^3+n+k} leq frac{n^2}{n^3+n}$$
It follows
$$ncdot frac{n^2}{n^3+2n} leq y_n leq n cdot frac{n^2}{n^3+n}$$
Now, take the limits on both sides.
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
add a comment |
up vote
4
down vote
accepted
Hint:
You may squeeze $y_n$ as follows:
- For $k=1, ldots , n$ you have
$$frac{n^2}{n^3+2n} leq frac{n^2}{n^3+n+k} leq frac{n^2}{n^3+n}$$
It follows
$$ncdot frac{n^2}{n^3+2n} leq y_n leq n cdot frac{n^2}{n^3+n}$$
Now, take the limits on both sides.
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Hint:
You may squeeze $y_n$ as follows:
- For $k=1, ldots , n$ you have
$$frac{n^2}{n^3+2n} leq frac{n^2}{n^3+n+k} leq frac{n^2}{n^3+n}$$
It follows
$$ncdot frac{n^2}{n^3+2n} leq y_n leq n cdot frac{n^2}{n^3+n}$$
Now, take the limits on both sides.
Hint:
You may squeeze $y_n$ as follows:
- For $k=1, ldots , n$ you have
$$frac{n^2}{n^3+2n} leq frac{n^2}{n^3+n+k} leq frac{n^2}{n^3+n}$$
It follows
$$ncdot frac{n^2}{n^3+2n} leq y_n leq n cdot frac{n^2}{n^3+n}$$
Now, take the limits on both sides.
answered Nov 5 at 7:12
trancelocation
8,9051521
8,9051521
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
add a comment |
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
Thanks a lot. I got it too !
– Yadati Kiran
Nov 5 at 7:24
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
You are welcome. :-)
– trancelocation
Nov 5 at 7:26
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
Cauchy's first theorem fir limits can also be used.
– Devendra Singh Rana
Nov 13 at 17:16
add a comment |
up vote
1
down vote
Just added for your curiosity.
If you know harmonic numbers,
$$displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}=n^2 left(H_{n^3+2 n}-H_{n^3+n}right)$$ Now, using, for large $p$
$$H_p=gamma +log left({p}right)+frac{1}{2 p}-frac{1}{12
p^2}+Oleft(frac{1}{p^3}right)$$ and continuing with long division or better with Taylor series for large $n$, you should get
$$y_n=1-frac{3}{2 n^2}+Oleft(frac{1}{n^3}right)$$ For example
$$y_{10}=frac{6646021071305823047954845}{6748980404790313121758326}approx 0.9847$$ while the above approximation would give $frac{197}{200}=0.9850$.
add a comment |
up vote
1
down vote
Just added for your curiosity.
If you know harmonic numbers,
$$displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}=n^2 left(H_{n^3+2 n}-H_{n^3+n}right)$$ Now, using, for large $p$
$$H_p=gamma +log left({p}right)+frac{1}{2 p}-frac{1}{12
p^2}+Oleft(frac{1}{p^3}right)$$ and continuing with long division or better with Taylor series for large $n$, you should get
$$y_n=1-frac{3}{2 n^2}+Oleft(frac{1}{n^3}right)$$ For example
$$y_{10}=frac{6646021071305823047954845}{6748980404790313121758326}approx 0.9847$$ while the above approximation would give $frac{197}{200}=0.9850$.
add a comment |
up vote
1
down vote
up vote
1
down vote
Just added for your curiosity.
If you know harmonic numbers,
$$displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}=n^2 left(H_{n^3+2 n}-H_{n^3+n}right)$$ Now, using, for large $p$
$$H_p=gamma +log left({p}right)+frac{1}{2 p}-frac{1}{12
p^2}+Oleft(frac{1}{p^3}right)$$ and continuing with long division or better with Taylor series for large $n$, you should get
$$y_n=1-frac{3}{2 n^2}+Oleft(frac{1}{n^3}right)$$ For example
$$y_{10}=frac{6646021071305823047954845}{6748980404790313121758326}approx 0.9847$$ while the above approximation would give $frac{197}{200}=0.9850$.
Just added for your curiosity.
If you know harmonic numbers,
$$displaystyle y_n=sum_{k=1}^n frac{n^2}{n^3+n+k}=n^2 left(H_{n^3+2 n}-H_{n^3+n}right)$$ Now, using, for large $p$
$$H_p=gamma +log left({p}right)+frac{1}{2 p}-frac{1}{12
p^2}+Oleft(frac{1}{p^3}right)$$ and continuing with long division or better with Taylor series for large $n$, you should get
$$y_n=1-frac{3}{2 n^2}+Oleft(frac{1}{n^3}right)$$ For example
$$y_{10}=frac{6646021071305823047954845}{6748980404790313121758326}approx 0.9847$$ while the above approximation would give $frac{197}{200}=0.9850$.
answered Nov 5 at 8:33
Claude Leibovici
118k1156131
118k1156131
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%2f2985377%2flimit-of-an-infinite-sum-y-n%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