Calculating $sumlimits_{n=1}^inftyfrac{{1}}{n+3^n} $
7
$begingroup$
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
$endgroup$
|
show 11 more comments
7
$begingroup$
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
$endgroup$
2
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
2
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
1
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
1
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
3
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14
|
show 11 more comments
7
7
7
4
$begingroup$
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
$endgroup$
I was able to prove this sum
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
calculus sequences-and-series convergence summation
calculus sequences-and-series convergence summation
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
edited Dec 2 '18 at 0:51
David G. Stork
10.2k21332
10.2k21332
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
asked Dec 2 '18 at 0:05
Ford DavisFord Davis
443
443
2
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
2
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
1
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
1
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
3
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14
|
show 11 more comments
2
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
2
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
1
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
1
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
3
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14
2
2
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
2
2
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
1
1
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
1
1
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
3
3
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14
|
show 11 more comments
1 Answer
1
active
oldest
votes
3
$begingroup$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
answered Dec 2 '18 at 2:24
gimusigimusi
1
$endgroup$
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022037%2fcalculating-sum-limits-n-1-infty-frac1n3n%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
3
$begingroup$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
answered Dec 2 '18 at 2:24
gimusigimusi
1
$endgroup$
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
add a comment |
3
$begingroup$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
answered Dec 2 '18 at 2:24
gimusigimusi
1
$endgroup$
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
add a comment |
3
3
3
$begingroup$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
answered Dec 2 '18 at 2:24
gimusigimusi
1
$endgroup$
What we can obtain, also by hand, is a reasonable estimation for example by
$$sum_{n=1}^inftyfrac{{1}}{n+3^n}approx 0.392<sum_{n=1}^3frac{{1}}{n+3^n}+sum_{n=4}^inftyfrac{{1}}{3^n}=$$$$=frac14+frac1{11}+frac1{30}+frac32-1-frac13-frac1{9}-frac1{27}approx 0.393$$
answered Dec 2 '18 at 2:24
gimusigimusi
1
edited Dec 2 '18 at 9:14
answered Dec 2 '18 at 2:24
gimusigimusi
1
answered Dec 2 '18 at 2:24
gimusigimusi
1
answered Dec 2 '18 at 2:24
gimusigimusi
1
1
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
add a comment |
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
I dont remember us going over approximating the sum, but this might be what she wanted. Thanks for your help.
$endgroup$
– Ford Davis
Dec 2 '18 at 2:40
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
$begingroup$
@FordDavis Yes indeed in that case seems convenient use the fact that $3^n$ is larger than $n$ also for $n$ small to obtain a god estimation for the series.
$endgroup$
– gimusi
Dec 2 '18 at 9:13
add a comment |
draft saved
draft discarded
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.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022037%2fcalculating-sum-limits-n-1-infty-frac1n3n%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
2
$begingroup$
@FordDavis Ask your teacher about the question again. It's quite unlikely they expect for you to find the sum. Or maybe the question was $frac{1}{n3^n}$, which has a much nicer answer.
$endgroup$
– Toby Mak
Dec 2 '18 at 0:21
2
$begingroup$
What about this one: Is there any non-zero function $f(n) $ such that $sum_{n=1}^infty frac{1} {f(n) +3^n}$ has an okay closed form?
$endgroup$
– Zacky
Dec 2 '18 at 0:22
1
$begingroup$
@Zacky There are some trivial choices like $f(n) = 2^n - 3^n$.
$endgroup$
– Daniel
Dec 2 '18 at 0:33
1
$begingroup$
There's something odd about this assignment. Part b) can be summed in closed form, but it takes techniques way beyond calc 2. Are you perhaps supposed to approximate the sums. I definitely second @TobyMak's advice: ask your teacher about it.
$endgroup$
– saulspatz
Dec 2 '18 at 0:33
3
$begingroup$
As noted by Carl, this question already exists ... math.stackexchange.com/q/1672121/442 but even that has no answer in nearly 3 years. No answer = cannot mark this as a duplicate.
$endgroup$
– GEdgar
Dec 2 '18 at 1:14