Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
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
2  
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
1  
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
1  
@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
1  
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 1:26
show 7 more comments

2 Answers

up vote 9 down vote accepted

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

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

--

$^1$ 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 $\frac{1}{2k}$.

share|improve this answer
    
I'd upvote this if I could but I don't have the necessary reputation just yet. –  Curious Jan 23 '12 at 11:54
    
@Curious: Thanks! –  Américo Tavares Jan 23 '12 at 15:11
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.