Find the limit of $left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)$ as $n to infty$.
Find the limit of $left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)$ as $n to infty$.
By expansion -
$$limlimits_{n to infty} left[1+(n^{2})(2/n) + (n^{2})(n^{2}-1)/2 dots ]/[1+2n+(2n)^{3}/3! dotsright]$$
I didn't get any result.
By applying limit directly, I'm getting indeterminate form.
How to find this limit?
limits
edited Jan 6 at 10:40
rtybase
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
add a comment |
Find the limit of $left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)$ as $n to infty$.
By expansion -
$$limlimits_{n to infty} left[1+(n^{2})(2/n) + (n^{2})(n^{2}-1)/2 dots ]/[1+2n+(2n)^{3}/3! dotsright]$$
I didn't get any result.
By applying limit directly, I'm getting indeterminate form.
How to find this limit?
limits
edited Jan 6 at 10:40
rtybase
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
add a comment |
Find the limit of $left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)$ as $n to infty$.
By expansion -
$$limlimits_{n to infty} left[1+(n^{2})(2/n) + (n^{2})(n^{2}-1)/2 dots ]/[1+2n+(2n)^{3}/3! dotsright]$$
I didn't get any result.
By applying limit directly, I'm getting indeterminate form.
How to find this limit?
limits
edited Jan 6 at 10:40
rtybase
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
Find the limit of $left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)$ as $n to infty$.
By expansion -
$$limlimits_{n to infty} left[1+(n^{2})(2/n) + (n^{2})(n^{2}-1)/2 dots ]/[1+2n+(2n)^{3}/3! dotsright]$$
I didn't get any result.
By applying limit directly, I'm getting indeterminate form.
How to find this limit?
limits
limits
edited Jan 6 at 10:40
rtybase
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
edited Jan 6 at 10:40
rtybase
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
edited Jan 6 at 10:40
rtybase
10.5k21533
edited Jan 6 at 10:40
rtybase
10.5k21533
edited Jan 6 at 10:40
rtybase
10.5k21533
10.5k21533
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
asked Jan 6 at 10:05
MathsaddictMathsaddict
2908
2908
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Hint. Note that
$$left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)=expleft(n^2logleft(1+ frac{2}{n}right)-2nright).$$
Now, by using the expansion $log(1+t)=t-frac{t^2}{2}+o(t^2)$ at $t=0$, we have that
$$logleft(1+ frac{2}{n}right)=frac{2}{n}-frac{2}{n^2}+o(1/n^2).$$
Can you take it from here?
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
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%2f3063674%2ffind-the-limit-of-left1-frac2n-rightn2-exp-2n-as-n-to-inf%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. Note that
$$left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)=expleft(n^2logleft(1+ frac{2}{n}right)-2nright).$$
Now, by using the expansion $log(1+t)=t-frac{t^2}{2}+o(t^2)$ at $t=0$, we have that
$$logleft(1+ frac{2}{n}right)=frac{2}{n}-frac{2}{n^2}+o(1/n^2).$$
Can you take it from here?
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
add a comment |
Hint. Note that
$$left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)=expleft(n^2logleft(1+ frac{2}{n}right)-2nright).$$
Now, by using the expansion $log(1+t)=t-frac{t^2}{2}+o(t^2)$ at $t=0$, we have that
$$logleft(1+ frac{2}{n}right)=frac{2}{n}-frac{2}{n^2}+o(1/n^2).$$
Can you take it from here?
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
add a comment |
Hint. Note that
$$left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)=expleft(n^2logleft(1+ frac{2}{n}right)-2nright).$$
Now, by using the expansion $log(1+t)=t-frac{t^2}{2}+o(t^2)$ at $t=0$, we have that
$$logleft(1+ frac{2}{n}right)=frac{2}{n}-frac{2}{n^2}+o(1/n^2).$$
Can you take it from here?
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
Hint. Note that
$$left(1+ frac{2}{n}right)^{n^{2}} exp(-2n)=expleft(n^2logleft(1+ frac{2}{n}right)-2nright).$$
Now, by using the expansion $log(1+t)=t-frac{t^2}{2}+o(t^2)$ at $t=0$, we have that
$$logleft(1+ frac{2}{n}right)=frac{2}{n}-frac{2}{n^2}+o(1/n^2).$$
Can you take it from here?
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
edited Jan 6 at 10:15
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
answered Jan 6 at 10:07
Robert ZRobert Z
93.8k1061132
93.8k1061132
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
add a comment |
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
1
1
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
Okay, I got exp(-2). Thanks!
– Mathsaddict
Jan 6 at 10:19
1
1
Yes, that's it!
– Robert Z
Jan 6 at 10:34
Yes, that's it!
– Robert Z
Jan 6 at 10:34
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
maybe $O(1/n^3)$?
– John Joy
Jan 6 at 11:08
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
@JohnJoy Yes, here we can replace $o(1/n^2)$ with $O(1/n^3)$, but $o(1/n^2)$ suffices.
– Robert Z
Jan 6 at 13:31
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%2f3063674%2ffind-the-limit-of-left1-frac2n-rightn2-exp-2n-as-n-to-inf%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
