# proof of convergence criterion for infinite product

Consider the partial product^{} ${P}_{n}={\prod}_{i=1}^{n}{p}_{i}$.

By definition we say that the infinite product ${\prod}_{n=1}^{\mathrm{\infty}}{p}_{n}$ is convergent iff ${P}_{n}$ is convergent.

Suppose every ${p}_{n}>0$

$\mathrm{ln}$ is a continuous^{} bijection from ${\mathbb{R}}^{+}$ to $\mathbb{R}$, therefore
${lim}_{n\to \mathrm{\infty}}{a}_{n}=a\iff {lim}_{n\to \mathrm{\infty}}\mathrm{ln}({a}_{n})=\mathrm{ln}(a)$, provided ${a}_{n}>0$ and $a>0$.

so saying ${P}_{n}\to P>0$ is equivalent^{} to saying that $\mathrm{ln}({P}_{n})$ converges^{}.

Since $\mathrm{ln}({P}_{n})=\mathrm{ln}({\prod}_{i=1}^{n}{p}_{i})={\sum}_{i=1}^{n}\mathrm{ln}({p}_{i})$, the infinite product converges to a positive value iff the series ${\sum}_{n=1}^{\mathrm{\infty}}\mathrm{ln}({p}_{n})$ is convergent.

In particular, if the infinite product converges to a positive value, then $\mathrm{ln}({p}_{n})\to 0\u27f9{p}_{n}\to 1$.

${P}_{n}\to 0$, is equivalent to saying ${\sum}_{n=1}^{\mathrm{\infty}}\mathrm{ln}({p}_{n})=-\mathrm{\infty}$

For the second part of the theorem:

${\prod}_{n=1}^{\mathrm{\infty}}(1+{p}_{n})$ converges absolutely to a positive value iff ${\sum}_{n=1}^{\mathrm{\infty}}{p}_{n}$ converges absolutely.

as we have seen, $1+{p}_{n}\to 1\u27f9{p}_{n}\to 0$

consider: ${lim}_{x\to 0}\frac{\mathrm{ln}(1+x)}{x}=1$ (this is easy to prove since by Taylor’s expansion $\mathrm{ln}(1+x)=x+O({x}^{2})$)

Since ${p}_{n}\to 0$ we can say that ${lim}_{n\to \mathrm{\infty}}\frac{\mathrm{ln}(1+{p}_{n})}{{p}_{n}}=1$ and by the limit comparison test^{}, either both ${\sum}_{n=1}^{\mathrm{\infty}}\mathrm{ln}(1+{p}_{n})$ and ${\sum}_{i=1}^{n}{p}_{i}$ converge or diverge.

Title | proof of convergence criterion for infinite product |
---|---|

Canonical name | ProofOfConvergenceCriterionForInfiniteProduct |

Date of creation | 2013-03-22 15:35:36 |

Last modified on | 2013-03-22 15:35:36 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 10 |

Author | cvalente (11260) |

Entry type | Proof |

Classification | msc 26E99 |