$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$ converges or diverges?












1












$begingroup$


How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?



When $n$ is odd, series is $sum frac{-1}{2n}$



When $n$ is even, series is $sum frac{3}{2n}$



This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$



$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.



Is this method right?
Please, suggest if there is some easier way.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:17


















1












$begingroup$


How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?



When $n$ is odd, series is $sum frac{-1}{2n}$



When $n$ is even, series is $sum frac{3}{2n}$



This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$



$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.



Is this method right?
Please, suggest if there is some easier way.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:17
















1












1








1


0



$begingroup$


How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?



When $n$ is odd, series is $sum frac{-1}{2n}$



When $n$ is even, series is $sum frac{3}{2n}$



This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$



$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.



Is this method right?
Please, suggest if there is some easier way.










share|cite|improve this question









$endgroup$




How to check if the series $$sum_{n=1}^{infty} frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?



When $n$ is odd, series is $sum frac{-1}{2n}$



When $n$ is even, series is $sum frac{3}{2n}$



This series is similar to the series
$$sum frac{-1}{2(2n-1)} + frac{3}{2(2n)}$$



$$= sum frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.



Is this method right?
Please, suggest if there is some easier way.







sequences-and-series






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 26 '18 at 13:10









MathsaddictMathsaddict

3619




3619












  • $begingroup$
    Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:17




















  • $begingroup$
    Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:17


















$begingroup$
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
$endgroup$
– hamam_Abdallah
Dec 26 '18 at 13:17






$begingroup$
Your method is not correct. You distinguish $n$ even and $n$ odd for the partial sum not for the general term.
$endgroup$
– hamam_Abdallah
Dec 26 '18 at 13:17












2 Answers
2






active

oldest

votes


















2












$begingroup$

hint



$$sum frac{(-1)^n}{n}$$



is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.



The sum of a convergent and a divergent series is a Divergent one.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    The sum should diverge. Right?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:24










  • $begingroup$
    @Mathsaddict Yes of course. but Div +Div could converge.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:25



















3












$begingroup$

The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    From my computations, sequence of partiat sum diverges, and so does the series.
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:29










  • $begingroup$
    You write as if that's different from what I wrote.
    $endgroup$
    – José Carlos Santos
    Dec 26 '18 at 13:41










  • $begingroup$
    I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:52













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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052925%2fsum-n-1-infty-frac1-2-1nn-converges-or-diverges%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









2












$begingroup$

hint



$$sum frac{(-1)^n}{n}$$



is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.



The sum of a convergent and a divergent series is a Divergent one.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    The sum should diverge. Right?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:24










  • $begingroup$
    @Mathsaddict Yes of course. but Div +Div could converge.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:25
















2












$begingroup$

hint



$$sum frac{(-1)^n}{n}$$



is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.



The sum of a convergent and a divergent series is a Divergent one.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    The sum should diverge. Right?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:24










  • $begingroup$
    @Mathsaddict Yes of course. but Div +Div could converge.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:25














2












2








2





$begingroup$

hint



$$sum frac{(-1)^n}{n}$$



is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.



The sum of a convergent and a divergent series is a Divergent one.






share|cite|improve this answer











$endgroup$



hint



$$sum frac{(-1)^n}{n}$$



is convergent by alternate criteria.
$$frac 12sum frac 1n$$
is a divergent Riemann series.



The sum of a convergent and a divergent series is a Divergent one.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 26 '18 at 13:42

























answered Dec 26 '18 at 13:14









hamam_Abdallahhamam_Abdallah

38.1k21634




38.1k21634












  • $begingroup$
    The sum should diverge. Right?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:24










  • $begingroup$
    @Mathsaddict Yes of course. but Div +Div could converge.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:25


















  • $begingroup$
    The sum should diverge. Right?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:24










  • $begingroup$
    @Mathsaddict Yes of course. but Div +Div could converge.
    $endgroup$
    – hamam_Abdallah
    Dec 26 '18 at 13:25
















$begingroup$
The sum should diverge. Right?
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:24




$begingroup$
The sum should diverge. Right?
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:24












$begingroup$
@Mathsaddict Yes of course. but Div +Div could converge.
$endgroup$
– hamam_Abdallah
Dec 26 '18 at 13:25




$begingroup$
@Mathsaddict Yes of course. but Div +Div could converge.
$endgroup$
– hamam_Abdallah
Dec 26 '18 at 13:25











3












$begingroup$

The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    From my computations, sequence of partiat sum diverges, and so does the series.
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:29










  • $begingroup$
    You write as if that's different from what I wrote.
    $endgroup$
    – José Carlos Santos
    Dec 26 '18 at 13:41










  • $begingroup$
    I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:52


















3












$begingroup$

The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    From my computations, sequence of partiat sum diverges, and so does the series.
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:29










  • $begingroup$
    You write as if that's different from what I wrote.
    $endgroup$
    – José Carlos Santos
    Dec 26 '18 at 13:41










  • $begingroup$
    I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:52
















3












3








3





$begingroup$

The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.






share|cite|improve this answer









$endgroup$



The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$left(sum_{n=1}^Nfrac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges. If it does, then the sequence$$left(sum_{n=1}^{2N}frac{frac12+(-1)^n}nright)_{Ninmathbb N}$$converges too. But it follows from your computations that it doesn't.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 26 '18 at 13:16









José Carlos SantosJosé Carlos Santos

163k22131234




163k22131234












  • $begingroup$
    From my computations, sequence of partiat sum diverges, and so does the series.
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:29










  • $begingroup$
    You write as if that's different from what I wrote.
    $endgroup$
    – José Carlos Santos
    Dec 26 '18 at 13:41










  • $begingroup$
    I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:52




















  • $begingroup$
    From my computations, sequence of partiat sum diverges, and so does the series.
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:29










  • $begingroup$
    You write as if that's different from what I wrote.
    $endgroup$
    – José Carlos Santos
    Dec 26 '18 at 13:41










  • $begingroup$
    I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
    $endgroup$
    – Mathsaddict
    Dec 26 '18 at 13:52


















$begingroup$
From my computations, sequence of partiat sum diverges, and so does the series.
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:29




$begingroup$
From my computations, sequence of partiat sum diverges, and so does the series.
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:29












$begingroup$
You write as if that's different from what I wrote.
$endgroup$
– José Carlos Santos
Dec 26 '18 at 13:41




$begingroup$
You write as if that's different from what I wrote.
$endgroup$
– José Carlos Santos
Dec 26 '18 at 13:41












$begingroup$
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:52






$begingroup$
I thought you are pointing towards some mistake in my calculations, since you wrote 'but' in the last line. Does your answer support my method to check if the series converges?
$endgroup$
– Mathsaddict
Dec 26 '18 at 13:52




















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














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052925%2fsum-n-1-infty-frac1-2-1nn-converges-or-diverges%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Ellipse (mathématiques)

Quarter-circle Tiles

Mont Emei