2

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln n$$ is convergent.

I wonder that whether there exist a formula (closed form) for the obtain value of the following finite summation. $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}$$ If not why?

Martin Sleziak
  • 53,687
Bumblebee
  • 18,220
  • 5
  • 47
  • 87
  • 8
    It's not at all clear how this series is defined, unless the fourth term should be $\frac{1}{4}$, in which case no such closed form is known. – J. J. Aug 28 '14 at 11:08
  • in addition to @J.J. comment: check out the Harmonic series – Alex Aug 28 '14 at 11:09
  • @J.J. Thanks. I know that $\sum_{k=1}^n \dfrac{1}{k}> \ln (n+1)$ and $\sum_{k=1}^{2^n} \dfrac{1}{k}> 1+ \dfrac{n}{2}.$ I wont to know that why there does not exist a such closed form. Have any one prove that there can not be a such closed form? – Bumblebee Aug 28 '14 at 11:17
  • 1
    @Nilan I think you are looking for the $n^{th}$ harmonic number, $H_n$ but in the future, it would be better for you to explicitly state your series. – Jam Aug 28 '14 at 11:17
  • @Nilan the sequence $\sum_{k=1}^{\infty} (1/k-ln(k))$ is divergent, too-not convergent. – callculus42 Aug 28 '14 at 11:29
  • @calculus Search for Euler Mascheroni_constant – Bumblebee Aug 28 '14 at 11:36
  • @Nilan Now it is clear, what you meant. – callculus42 Aug 28 '14 at 11:53

2 Answers2

6

$$H_n~=~\psi(n+1)+\gamma~=~-\int_0^1\ln\Big(1-\sqrt[n]x\Big)~dx~=~\int_0^1\frac{1-x^n}{1-x~~}~dx,$$ where $\psi$ is the digamma function, and $\gamma$ is the Euler-Mascheroni constant. See harmonic numbers for more details.

Lucian
  • 48,334
  • 2
  • 83
  • 154
1

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}=\sum_{i=1}^{n} \frac{1}{i}=H_n$$

where $H_n$ is the $n^{th}$ harmonic number.