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It is possible to deduce the value of the following (in my opinion) converging infinite series? If yes, then what is it?

$$\sum_{n=1}^\infty\frac{1}{2\cdot n}$$

where n is an integer. Sorry if the notation is a bit off, I hope youse get the idea.

jimjim
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Curious
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    The series diverges: it’s simply $1/2$ of the harmonic series, which is a standard example of a divergent series. – Brian M. Scott Jan 23 '12 at 11:38
  • The series diverges! Do you know that Harmonic Series Diverges? Then use that fact (it requires proof!) to conclude that this series diverges! –  Jan 23 '12 at 11:38
  • You'll have a hard time with that one. If you factor out the half, then you are left with the harmonic series. – Henry Shearman Jan 23 '12 at 11:39
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    Opinions aren't worth much in mathematics - do you have any reason for thinking that series converges? – Gerry Myerson Jan 23 '12 at 11:44
  • @GerryMyerson - As you might have guessed, I am not a mathematician and unfortunately I don't have the time to become one. That is why I asked. – Curious Jan 23 '12 at 11:50
  • @Curious Are you willing to know the proof of divergence ofHarmonic series! If you already know, I suggest that you upvote and accept Americo Tavares' Answer as it addresses your problem. The question will stand solved for all purposes in the site! –  Jan 23 '12 at 11:54
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    OK, but usually when people have opinions, those opinions are based on something. I'm "curious" as to the basis for your opinion that the series converges. – Gerry Myerson Jan 23 '12 at 12:00
  • @KannappanSampath - Yes I am willing to accept the official position of Your church. So be it, bless You with Euler, Fermat and other saints. – Curious Jan 23 '12 at 12:52
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    @GerryMyerson - and surely the basis of an opinion must be based on something else etc etc. But seriously.. I got a decent answer and see no benefit in arguing over supposed merits or shortcomings of reasoning that lead to this question. Thanks again, and take care. – Curious Jan 23 '12 at 13:14
  • See also this question: http://math.stackexchange.com/questions/255/why-does-the-series-frac-1-1-frac-12-frac-13-cdots-not-converge – Martin Sleziak Jan 23 '12 at 13:51
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    I'm glad you got a helpful answer. The benefit in discussing the reasoning that led to a question is that you might learn something about how to decide mathematical questions that interest you, or how to phrase such questions when asking them of other people. I don't see the benefit in shutting down discussion of unresolved points. – Gerry Myerson Jan 24 '12 at 01:26

2 Answers2

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The series is not convergent, since it is half of the harmonic series which is known to be divergent$^1$.

$$\sum_{n=1}^{\infty }\frac{1}{2n}=\frac{1}{2}\sum_{n=1}^{\infty }\frac{1}{n}.$$

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$^1$ The sum of the following $k$ terms is greater or equal to $\frac{1}{2}$

$$\frac{1}{k+1}+\frac{1}{k+2}+\ldots +\frac{1}{2k-1}+\frac{1}{2k}\geq k\times \frac{1}{2k}=\frac{1}{2},$$

because each term is greater or equal to $\frac{1}{2k}$.

Américo Tavares
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To me, the easiest way to see that the harmonic series diverges is to use the Integral test. Then you do not have to deal with coming up with a formula.

toypajme
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