# proof of infinite product of sums $1\text{tmspace}-.1667em+\text{tmspace}-.1667em{a}_{i}$ result without exponentials

In this entry, we show how the proof presented in the parent entry
may be modified so as to avoid use of the exponential function^{}.
This modification makes it more elementary by not requiring that
one first develop the theory of the exponential function before
proving this basic result about infinite products. Note that it
is only necessary to redo the part of the result which states that,
if the series converges, then the product also converges because
the proof of the opposite implication did not involve the exponential
function.

We begin with a simple inequality^{}. Suppose that $a$ and $b$ are
real numbers such that $0\le a$ and $$. Then we
have $2ab\le a$, hence

$(1+a)(1+2b)$ | $=1+2b+a+2ab$ | ||

$\le 1+2b+a+a$ | |||

$=1+2(a+b).$ |

Now suppose that the series ${a}_{1}+{a}_{2}+{a}_{3}+\mathrm{\cdots}$ converges to a value $S$. Since the convergence of an infinite series or product is not affected by removing a finite number of terms we may, without loss of generality, assume that $$. Then, since the terms ${a}_{n}$ are nonnegative for all $n$, for each partial sum ${s}_{n}$ we will have $$.

Clearly, ${t}_{1}\le 1+2{s}_{1}$. Suppose that, for some $n$, we have ${t}_{n}\le 1+2{s}_{n}$. Then, using the definitions of ${t}_{n}$ and ${s}_{n}$ along with the inequality demonstrated above, we conclude that

${t}_{n+1}$ | $={t}_{n}(1+{a}_{n+1})$ | ||

$\le (1+2{s}_{n})(1+{a}_{n+1})$ | |||

$\le 1+2({s}_{n}+{a}_{n+1})$ | |||

$=1+2{s}_{n+1}$ |

Hence, if ${t}_{n}\le 1+2{s}_{n}$, then ${t}_{n+1}\le 1+2{s}_{n+1}$ as well. By induction, we conclude that ${t}_{n}\le 1+2{s}_{n}$ for all $n$.

Thus, for all $n$, we have ${t}_{n}\le 1+2{s}_{n}\le 1+2S$. Substituting this inequality for the inequality ${t}_{n}\le {e}^{{s}_{n}}\le {e}^{S}$ in the parent entry, the rest of the proof proceeds in exactly the same manner.

Title | proof of infinite product of sums $1\text{tmspace}-.1667em+\text{tmspace}-.1667em{a}_{i}$ result without exponentials^{} |
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Canonical name | ProofOfInfiniteProductOfSums1aiResultWithoutExponentials |

Date of creation | 2013-03-22 18:40:38 |

Last modified on | 2013-03-22 18:40:38 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Result |

Classification | msc 40A20 |

Classification | msc 26E99 |