Different values?

Number Theory Level 3

Define an infinite series as follows

$S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}...$

If you were able to permute and rearrange this series in whatever way you want, what can $S$ converge to?

Only one value Infinitely many only positive rational values Any value in the set of all real numbers Infinitely many only irrational values Infinitely many only positive values

1 solution

Jacopo Piccione
Dec 27, 2018

Riemann rearrangement theorem says that if a infinite series is convergent, but not absolutely convergent, its terms can be rearranged in a permutation so that the new series converges to an arbitrary real number or diverges.

The series is convergent: it's the alternating harmonic series, which it's known to converge to $ln2$ .

But it's not absolutely convergent: $1+\frac{1}{2} +\frac{1}{3}+\frac{1}{4}+...$ is the harmonic series, which diverges.

So, for the Riemann rearrangement theorem, a permutation of $S$ can converge to an arbitrary real number.

Different values?

1 solution

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