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Can someone give a simple explanation for why the harmonic series

$$\frac 1 1 + \frac 12 + \frac 13 + \cdots $$

doesn't converge, but just grows very slowly?

I'd prefer an easily comprehensible explanation rather than a rigorous proof of the type I could get from an undergraduate text book.

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This is not meant to be an answer but an interesting note. Suppose we denote $H(n) = 1/1 + 1/2 + ... + 1/n$ then $H(n!) - H((n-1)!) \approx log(n)$ for large n. Does this give a hint? ;) –  Roupam Ghosh Jul 11 '11 at 4:14
    
Here is a weakly related question: What is a textbook, or even a popularization for the general public, that (1) discusses infinite series, but (2) does not have an explanation for the divergence of this exact series? –  GEdgar Nov 3 '13 at 19:50
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15 Answers

up vote 67 down vote accepted

Let's group the terms as follows:

group 1: 1/1 (1 term)

group 2: 1/2 + 1/3 (2 terms)

group 3: 1/4 + 1/5 + 1/6 + 1/7 (4 terms)

group 4: 1/8 + 1/9 + ... + 1/15 (8 terms)

$\vdots$

In general, group $n$ contains $2^{n-1}$ terms. But also, notice that the smallest element in group n is larger than $1/2^n$. For example all elements in group 2 are larger than $1/2^2$. So the sum of the terms in each group is larger than $2^{n-1} \cdot (1/ 2^n) = 1/2$. Since there are infinitely many groups, and the sum in each group is larger than $1/2$, it follows that the total sum is infinite.

This proof is often attributed to Nicole Oresme.

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+1: This is nice: it's easy to turn this into a rigorous proof, and it even gives you a lower bound for the order of growth! –  Simon Nickerson Jul 21 '10 at 5:19
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I assume you mean that group 4 as 8 terms? Or do you mean to go all the way to 1/23? –  Tomas Lycken Jul 21 '10 at 7:37
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Is there a closed-form function for this value? –  John Gietzen Jul 21 '10 at 18:29
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Interestingly, this proof goes as far back as Nicole Oresme in the 14th century. Wikipedia has a nice display of this proof [en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29] –  Neil Mayhew Jul 22 '10 at 13:20
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@John: There's no explicit closed-form, but they're generally known as the Harmonic Numbers; there are a number of identities involving them (how to sum them or sum multiples of them, etc.) –  Steven Stadnicki Jul 10 '11 at 21:23
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There is a fantastic collection of $20$ different proofs that this series diverges. I recommend you read it (it can be found here). I especially like proof $14$, which appeals to triangular numbers for a sort of cameo role.


EDIT

It seems the original link is broken, due to the author moving to his own site. So I followed up and found the new link. In addition, the author has an extended addendum, bringing the total number of proofs to 42+.

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Proof 6 is also nice. –  leonbloy Mar 15 '13 at 20:51
    
+1, for the fantastic link!. –  boywholived Apr 23 '13 at 10:22
    
Apparently, the list has been updated. –  David Mitra Jun 19 '13 at 16:15
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The link is broken. –  Prism Oct 24 '13 at 2:45
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The answer given by AgCl is a classic one. And possibly pedagogically best; I don't know.

I also like the following argument. I'm not sure what students who are new to the topic will think about it.

Suppose 1 + 1/2 + 1/3 + 1/4 + ... adds up to some finite total S. Now group terms in the following way:

$$1 + \frac{1}{2} > \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$$

$$\frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$$

$$\frac{1}{5} + \frac{1}{6} > \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

Continuing in this way, we get $S > S$, a contradiction.

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Not really. From $S_n > T_n$ you can only conclude that $\lim S_n \ge \lim T_n$. –  lhf Jul 10 '11 at 21:24
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@lhf: That's right, but that can be easily fixed here (with $S_n = 1 + 1/2 + \dots + 1/2n$ and $T_n = 1 + 1/2 + \dots + 1/n$): we can use a better inequality, like say $S_n \ge T_n + 1/2$ (using just the first step) to conclude that $\lim S_n \ge \lim T_n + 1/2$, contradicting $S = \lim S_n = \lim T_n$. –  ShreevatsaR Jul 11 '11 at 4:18
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This is not as good an answer as AgCl's, nonetheless people may find it interesting.

If you're used to calculus then you might notice that the sum $$ 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$$ is very close to the integral from $1$ to $n$ of $\frac{1}{x}$. This definite integral is ln(n), so you should expect $1+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{n}$ to grow like $\ln(n)$.

Although this argument can be made rigorous, it's still unsatisfying because it depends on the fact that the derivative of $\ln(x)$ is $\frac{1}{x}$, which is probably harder than the original question. Nonetheless it does illustrate a good general heuristic for quickly determining how sums behave if you already know calculus.

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If you look at a Riemann sum for intervals with width 1, you can pretty quickly see that the integral of 1/x from 1 to infinity must be less than the sum of the harmonic series. –  Isaac Jul 21 '10 at 5:51
    
Thank you for adding this answer. I was hoping to avoid an answer that involved integration, so I also prefer AgCl's answer. But I am happy to see more than one demonstration/proof. –  bryn Jul 22 '10 at 11:33
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An alternative proof (translated and adapted from this comment by Filipe Oliveira, in Portuguese, posted also here). Let $ f(x)=\ln(1+x)$. Then $f'(x)=\dfrac {1}{1+x}$ and $ f'(0)=1$. Hence

$$\displaystyle\lim_{x\to 0}\dfrac{\ln(1+x)}{x}=\lim_{x\to 0}\dfrac{\ln(1+x)-\ln(1)}{x-0}=1,$$

and

$$ \displaystyle\lim_{n\to\infty} \dfrac{\ln\left(1+\dfrac{1}{n}\right)}{\dfrac {1}{n}}=1>0.$$

So, the series $\displaystyle\sum\dfrac{1}{n}$ and $\displaystyle\sum\ln\left(1+\dfrac {1}{n}\right)$ are both convergent or divergent. Since

$$\ln\left(1+\dfrac {1}{n}\right)=\ln\left(\dfrac{n+1}{n}\right)=\ln (n+1)-\ln(n),$$

we have

$$\displaystyle\sum_{n=1}^N\ln\left(1+\dfrac {1}{n}\right)=\ln(N+1)-\ln(1)=\ln(N+1).$$

Thus $\displaystyle\sum_{n=1}^{\infty}\ln\left(1+\dfrac {1}{n}\right)$ is divergent and so is $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n}$.

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Another interesting proof is based upon one of the consequences of the Lagrange's theorem applied on $\ln(x)$ function, namely:

$$\frac{1}{k+1} < \ln(k+1)-\ln(k)<\frac{1}{k} \space , \space k\in\mathbb{N} ,\space k>0$$

Taking $k=1,2,...,n$ values to the inequality and then summing all relations, we get the required result.

The proof is complete.

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There also exists a proof for the divergence of the harmonic series that involves the Integral Test. It goes as follows.

It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral. Specifically, consider the arrangement of rectangles shown in the figure of $ y = \dfrac {1}{x} $:

y=1/x

Each rectangle is $1$ unit wide and $\frac{1}{n}$ units high, so the total area of the rectangles is the sum of the harmonic series: $$ \displaystyle\sum \left( \text {enclosed rectangle are} \right) = \displaystyle\sum_{k=1}^{\infty} \dfrac {1}{k}. $$Now, the total area under the curve is given by $$ \displaystyle\sum_{1}^{\infty} \dfrac {\mathrm{d}x}{x} = \infty. $$Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, this proves that $$ \displaystyle\sum_{n=1}^{k} \dfrac {1}{n} > \displaystyle\sum_{1}^{k+1} \dfrac {\mathrm{d}x}{x} = \ln (k+1). $$This is the backbone of what we know today as the integral test.

Interestingly, the alternating harmonic series does converge: $$ \displaystyle\sum_{n=1}^{\infty} \dfrac {(-1)^n}{n} = \ln 2. $$And so does the $p$-harmonic series with $p>1$.

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enter image description here first suppose a=1+1/2+1/3+1/4+... converge show that a>a that's paradox

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Ideally use Latex. –  Alyosha Nov 1 '13 at 17:13
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enter image description here a=1+1/2+1/3+1/4+.... by grouping numbers show that a is diverge

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This was bumped, so I'll add a proof sweet proof I saw in this site. Exponentiate $H_n$ and get $$e^{H_n}=\prod_{k=1}^n e^{1/k}\gt\prod_{k=1}^n\left(1+\frac{1}{k}\right)=n+1.$$ Therefore, $H_n\gt\log(n+1)$, so we are done. We used $e^x\gt1+x$ and telescoped the resulting product.

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$$\int_{0}^{\infty}e^{-nx}dx=\frac1n$$

$$\sum_{n=1}^{\infty}\int_{0}^{\infty}e^{-nx}dx=\sum_{n=1}^{\infty}\frac1n$$

using the law of Geometric series

$$\int_{0}^{\infty}(\frac{1}{1-e^{-x}}-1)dx=H_n:n\to \infty$$

$$H_n:n\to \infty=\left [ \ln(e^x-1)-x \right ]_0^{\infty}\to\infty$$

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Another (different) answer, by the Cauchy Condensation Test :

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

The latter is obviously divergent, therefore the former diverges. This is THE shortest proof there is.

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I think the integral test gives the most intuitive explanation. Observe that $\int^n_1 1/x dx= log n$ The sum $\sum^n_{k=1}1/k$ can be viewed as the area of $n$ rectangles of height $1/k$, width $1$ (with the first one having it's left hand side on the y axis, and all having their bottom on the x axis). The graph of $x\mapsto 1/x$ can be drawn under these, so the sum will grow with $n$ at (least) as fast as the integral - hence will grow (at least) logarithmically.

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use \log to get nice formatting for $\log$ –  Tyler Nov 1 '13 at 17:24
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A non-rigorous explanation I thought of once: consider a savings scheme where you put a dollar in your piggy bank every day. So after $n$ days, you have $n$ dollars; clearly, your savings approach infinity. On the other hand, each day you add an additional $1/n$ proportion of your existing savings, "so" (the non-rigorous step) the accumulated percentage after $n$ days is $1 + 1/2 + \cdots + 1/n$.

This can be made rigorous through the infinite product argument $$\prod_{n = 1}^\infty (1 + \tfrac{1}{n}) < \infty \iff \sum_{n = 1}^\infty \frac{1}{n} < \infty$$ which is obtained, essentially, by taking the logarithm of the left-hand side and using the power series for $\log (1 + x)$.

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Suppose to the contrary that converges.

Let $s_n$ denote the $n$-th partial sum. Since the serie converges, $(s_n)$ is a Cauchy sequence. Let $\varepsilon = 1/3$, then there is some $n_0$ such that $|s_q-s_p|< 1/3$ for all $q>p\ge n_0$. Let $q=2n_0$ and $p=n_0$. Then

$$\frac{1}{3}>\bigg|\sum_{n=n_0+1}^{2n_0} \frac{1}{n}\bigg|\ge\bigg|\sum_{n=n_0+1}^{2n_0} \frac{1}{2n_0}\bigg|=\frac{1}{2}$$

a contradiction. Then this contradiction shows that the series diverges.

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