Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.

— Niels Henrik Abel, letter to Holmboe (January 1826)

In 1665 the young Newton discovered that the binomial formula also held for exponents that are not whole numbers:

$$(1+x)^m=1+mx+\frac{m(m-1)}{2}x^2+\frac{m(m-1)(m-2)}{6}x^3+\cdots$$

For a positive integer $m$ the sum stops after $m+1$ terms. For $m=\frac12$ or $m=-1$ it never stops: it is a series, a sum with infinitely many terms. Newton, Euler and all the mathematicians of the eighteenth century used this series with enormous success. Almost nobody asked when it works.

Abel met this series while still at school. At 16 he noticed that the proof Euler had given of the binomial formula only held for rational exponents, and he tried to extend it to all numbers. Eight years later, in Berlin, he came back to the subject with far more rigour.

When the series deceives

Take $m=-1$. The series gives

$$\frac1{1+x}=1-x+x^2-x^3+x^4-\cdots$$

With $x=\frac12$, the left side is $\frac23$, and the right side, $1-0.5+0.25-0.125+\cdots$, does approach $0.666\ldots$ But with $x=1$ the right side becomes $1-1+1-1+\cdots$, which keeps jumping between $1$ and $0$ without making up its mind, while the left side is $\frac12$. Euler accepted that the sum “was” $\frac12$. And with $x=2$ it gets worse: the left side is $\frac13$, and the right side, $1-2+4-8+16-\cdots$, grows without limit.

A series whose partial sums do not approach any number is called divergent. Eighteenth-century mathematicians handled such series freely and sometimes reached absurd results. In France, Cauchy began putting the house in order in 1821, with his Course of analysis. Abel read Cauchy in Paris and went further.

Abel’s proof

In the 1826 paper in Crelle’s Journal, Abel studied the binomial series with full rigour, for any exponent $m$, even complex, and for any $x$. He proved that it converges to $(1+x)^m$ when $|x|<1$, and described precisely what happens when $|x|=1$, at the edge. It was the first time anyone had done this for Newton’s series.

Inside the interval the series works very well. With $m=\frac12$ the coefficients are $1,\ \frac12,\ -\frac18,\ \frac1{16},\ -\frac5{128},\dots$, and

$$\sqrt{1+x}=1+\frac x2-\frac{x^2}8+\frac{x^3}{16}-\frac{5x^4}{128}+\cdots$$

For $\sqrt{1.2}$, with $x=0.2$:

$$1+0.1-0.005+0.0005-0.0000625\approx1.0954.$$

The true value is $1.09545\ldots$: five terms already give four correct decimal places.

Abel’s theorem

The same paper contains a result that now bears his name. Suppose a power series $a_0+a_1x+a_2x^2+\cdots$ converges at $x=1$. Then the value of the series at $x=1$ is the limit of its values as $x$ approaches $1$ from below:

$$\boxed{\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty a_n.}$$

It looks obvious, but it is not: it is exactly the kind of passage to the limit that earlier mathematicians made without thinking and that sometimes went wrong. With the theorem it is guaranteed. An example: for $|x|<1$,

$$\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\cdots$$

At $x=1$ the series converges, because the terms alternate in sign and decrease. By Abel’s theorem its value is $\ln2$:

$$1-\frac12+\frac13-\frac14+\cdots=\ln2\approx0.6931.$$
Exercise: use four terms of the series for √(1 + x) to compute √0.81.

With $x=-0.19$: $1-0.095-0.0045125-0.0004287\approx0.90006$. The exact value is $0.9$. The fifth term, $-\frac{5x^4}{128}\approx-0.00005$, brings the result to $0.90001$.

Exercise: why can the series 1 − x + x² − x³ + … not converge at x = 2?

The terms are $1,\ -2,\ 4,\ -8,\ 16,\dots$ and grow without limit. A series can only converge if its terms approach zero, because each partial sum differs from the previous one by exactly the last term. Here the partial sums are $1,\ -1,\ 3,\ -5,\ 11,\dots$ and move further and further apart. The formula $\frac1{1+x}$ is still worth $\frac13$ at $x=2$; it is the series that stops being valid.

Sources: Niels Henrik Abel, “Untersuchungen über die Reihe $1+\frac m1x+\frac{m(m-1)}{1\cdot2}x^2+\cdots$”, Journal für die reine und angewandte Mathematik 1 (1826) · Øystein Ore, Niels Henrik Abel: Mathematician Extraordinary (1957) · Abel biography, MacTutor.

All topics on Niels Henrik Abel · ← The fifth-degree equation: the formula that does not exist · Inverting the integral: elliptic functions →

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