The sum of series $1-1/2-1/4+1/3-1/6-1/8+1/5-....$?

Question

I know that the sum of series $1-1/2+1/3-1/4+1/5..= log 2.$ And we can see that by rearranging the terms of the series given in question, i would land on the series as above. so this should mean that the answer should be log2. but no, the answer is log2/2.

also notice that the Question series has a positive term after every two negative terms.Does this have anything to do with the answer?

This question is different from the series $1-1/2+1/3-1/4+...$ as sum of this series is log2. but the sum of asked series is log2/2.

You can't rearrange the terms if your series is not absolutely convergent. — , Jan 13 '17 at 19:39
@OpenBall agreed that i shouldn't rearrange terms, but how should i proceed in such case? — Parul, Jan 13 '17 at 19:42
its not given in the question. we'd hv to formulate on our own — Parul, Jan 13 '17 at 19:45
Possible duplicate of Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ — , Jan 13 '17 at 20:33

spaceisdarkgreen · Answer 1 · 2017-01-13T20:16:07.030

5

Ok, looks like you are adding the odd terms and subtracting the even terms of the harmonic series in the order odd, even, even, odd, even, even.

We can write each trio of terms as $$ \frac{1}{n}-\frac{1}{2n}-\frac{1}{2n+2}$$

so reexpress the series as $$ \sum_{\mbox{n odd}} \frac{1}{n}-\frac{1}{2n}-\frac{1}{2n+2}.$$ The first two terms can be combined to get$$ \sum_{\mbox{n odd}} \frac{1}{2n}-\frac{1}{2n+2}= \frac{1}{2}\sum_{\mbox{n odd}}\frac{1}{n}-\frac{1}{n+1} = \frac{1}{2}\left(\frac{1}{1} -\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\ldots\right)$$ As you mentioned in your question, the thing inside the parentheses is $\log(2),$ so the final answer is $\frac{1}{2}\log(2).$

edited Jan 13 '17 at 20:16

answered Jan 13 '17 at 20:01

spaceisdarkgreen

58,508

while evaluating summation 1/n-1/n+1 from n=1 to infinty, we get 1+1/n. this is not that is required. – Parul Jan 13 '17 at 20:22
@Parul Not sure what you're saying? It's over odd n? (Maybe you saw it before this was fixed) – spaceisdarkgreen Jan 14 '17 at 00:16
Yes.sorry. I saw this before it was fixed.its absolutely correct. – Parul Jan 14 '17 at 02:54

score 3 · Accepted Answer · answered Jan 13 '17 at 20:01

If you take the terms 3 at a time, the formula for the $n$th term appears to be:

$$\frac{1}{2n-1} - \frac{1}{4n-2}- \frac{1}{4n}.$$

If so we have the above

$$= \frac{1}{4n(2n-1)} = \frac{1}{2} \frac{1}{2n(2n-1)}= \frac{1}{2}\left(\frac{1}{2n-1}-\frac{1}{2n}\right).$$

And this last is $\frac{1}{2}$ of two consecutive terms of the alternating harmonic series. This makes your sum equal to $\frac{1}{2} \ln 2.$

The sum of series $1-1/2-1/4+1/3-1/6-1/8+1/5-....$?

2 Answers2