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Background

When I attempt questions in real analysis, frequently I encounter the following expression like harmonic series:

$$H_n=1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$$

To my surprise when I arrive at something like this ,the solution book does not give me a closed-form of this expression.

Question

So my question is a simple one. What can you think of for this series? And besides the fact that it does not converge is it related to any mathematics that you know? And most importantly, does it have a closed-form?

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marked as duplicate by user21820, MisterRiemann, José Carlos Santos real-analysis Dec 6 '18 at 10:52

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ It is divergent ..... $\endgroup$ – neelkanth Dec 6 '18 at 7:05
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    $\begingroup$ Look out for the Euler-Mascheroni constant $\gamma$ if you want to explore the mathematical content behind your question. The sum is asymptotically $\ln n + \gamma$. The error term can be analysed. $\endgroup$ – Mark Bennet Dec 6 '18 at 7:06
  • $\begingroup$ OP probably meant to say "does not converge," since it's like the most well-known fact about the series. $\endgroup$ – Eevee Trainer Dec 6 '18 at 7:06
  • $\begingroup$ @neelkanth the typo is corrected $\endgroup$ – hephaes Dec 6 '18 at 7:21
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The harmonic series diverges, which is somewhat surprising because each term tends to zero as $n\to\infty$. However, the problem is that each term does not tend to zero fast enough.

A function which generalises this series is the Riemann zeta function $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$ which converges for $\text{Re}(s)>1$. For example,

$$\zeta(2)=\frac{\pi^2}{6}.$$

Euler gave a general formula for $\zeta(2n)$ in terms of the Bernoulli numbers, $$\zeta(2n)=\frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!},$$ but no closed form is known yet for odd arguments.

It can also be analytically continued to $\mathbb{C}\setminus\{1\}$. There is a singularity (pole of order 1) at $s=1$.

Zeros of this function lie on the negative even integers (the trivial zeros), whereas the non trivial (purely complex) zeros are all known to lie in the critical strip. In fact it is hypothesised that they all lie on the critical line $s=1/2+it$, which you may know is called the Riemann Hypothesis. If true, this has big implications concerning the distribution of the primes, and many conjectured theorems in Analytic Number Theory, since $\zeta(s)$ is related to the primes by Euler's product formula, $$\zeta(s)=\prod_{p\text{ prime}}\left(1-p^{-s}\right)^{-1}.$$

Coming back to "$\zeta(1)$" for a moment, as mentioned in the other answer, it pops up in the definition of Euler's gamma constant,

$$\gamma=\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}-\log n\approx 0.57721566490...$$

It is not yet known whether $\gamma$ is irrational, unlike $\zeta(2n)$, and also $\zeta(3)$ which was proved to be irrational by Apéry.

It may also be of interest to note that $$H_n=\sum_{k=1}^n\frac{1}{k}=\Psi(n+1)+\gamma,$$ where $\Psi$ is the Digamma function.

Another remarkable "closed form" is given by

$$H_n = \frac{\binom{(n+1)!+n}{n}-1}{(n+1)!}-(n+1)\Biggl\lfloor \frac{\binom{(n+1)!+n}{n}-1}{(n+1)(n+1)!}\Biggr\rfloor,$$

as stated by @nczksv in the "duplicate" answer.

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  • $\begingroup$ A closed form for $\zeta(2n+1)$ you find e.g. here (Multiple Gamma function, Generalized Glaisher-Kinkelin constant, formula 2.13), or with Multiple sine function e.g.here . You can also look through the links in here. $\endgroup$ – user90369 Dec 6 '18 at 10:44
  • $\begingroup$ @user90369 that all depends on the definition of "closed form," which admittedly I did not state. As far as I know there is no known closed form in terms of "elementary functions," similar to that for $\zeta(2n)$. $\endgroup$ – Antinous Dec 6 '18 at 12:04
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If I remember correctly, there is a sort of closed form:

$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{n} = \sum_{k=1}^n \frac{1}{k} = \ln(n) + \gamma + \epsilon_n$$

$\epsilon_n$ here is an error constant proportional to $1/2n$, and thus $\epsilon_n \to 0$ as $n \to \infty$.

$\gamma$ denotes the Euler-Mascheroni constant, defined by the limiting difference between the natural logarithm and the harmonic series, i.e.

$$\gamma =\lim _{n\to \infty }\left(-\ln n+\sum _{k=1}^{n}{\frac {1}{k}} \right)$$

Granted, I don't think this is really in line with what you want because ... well, it's sort of like a self-referential thing. "Oh, the harmonic series is given by a function, an error constant, and this special constant that comes from the difference from the series and that other function under certain conditions." It's mostly a personal thing so it doesn't mesh quite well with me?

Beyond that I don't really have anything to offer - specifically with relations of the series to mathematics I know - that I wouldn't just be regurgitating from Wikipedia. Weirdly it hasn't popped up much in my coursework thus far.

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    $\begingroup$ I don't think it showed up in my courses much either, aside from a useful series for comparison tests for divergence of series, but this seems to be true for quite a bit of what I've since found interesting and have studied. A nice survey at a level mostly accessible to strong students having completed calculus 2 (U.S. level) is Gamma: Exploring Euler's Constant by Julian Havil (2003), which I still see in local Barnes & Noble bookstores, so it's fairly well known. (continued) $\endgroup$ – Dave L. Renfro Dec 6 '18 at 8:34
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    $\begingroup$ In the late 1990s I wrote a handout on gamma for my (very strong high school) students several years before Havil's book came out -- at the time there wasn't anything easily available that I could share with my students about how slowly the harmonic series diverged. I had thought of expanding it some day, but then Havil's book appeared. I added Havil's book to the references, but otherwise haven't changed it from 1999. (continued) $\endgroup$ – Dave L. Renfro Dec 6 '18 at 8:42
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    $\begingroup$ A couple of more recent, and more technical papers are A survey of Euler's constant by Thomas Paul Dence and Joseph Bernard Dence (2009) and Euler's constant: Euler's work and modern developments by Jeffrey Clark Lagarias. The paper by Lagarias is very thorough, but it's also pitched at a much higher level than Havil's book and the paper by Dence/Dence. $\endgroup$ – Dave L. Renfro Dec 6 '18 at 8:47
  • $\begingroup$ Yeah, that's mostly what I recall using $1/n$ sort of stuff for, be it as an integral, a series or sequence, whatever - it's very convenient and simple to work with. I remember using the sequence $x_n = 1/n$ as a way of making examples and such in real analysis particularly. Seems weird that, for how handy it is in these respects, we don't actually study its behavior in the courses much. And thanks for the link to the book, I might take a look someday - a hefty 300 pages, that should really have a lot of goodies. And for the handouts and papers too. ^_^ $\endgroup$ – Eevee Trainer Dec 6 '18 at 8:50
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It diverges because, if you look at the partial sums, $s_{2n}-s_n=\frac1{2n}+\frac1{2n-1}+\cdots+\frac1{n+1}\ge n\cdot \frac1{2n}=\frac12\,,\forall n$. Thus it isn't Cauchy.

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