Divergence of a series

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March 20th 2011, 04:01 PM
Bruno J.

Divergence of a series

Here's a challenge problem I just made up:

Let $\{a_n\}$ be a sequence of positive real numbers such that $\sum a_n = \infty$ . Show that

$\frac{a_1}{a_0}+\frac{a_2}{a_0+a_1}+\frac{a_3}{a_0 +a_1+a_2}+ \dots = \infty$ .
March 20th 2011, 05:16 PM
PaulRS

Let $S_n = \sum_{k=0}^{n}a_k$ .

We want to show that $\sum_{k=0}^n\frac{\Delta S_k}{S_k}}$ diverges as $n\to \infty$ .
( I chose this notation to suggest the similarity with $\int_{ [ a,\infty)} \tfrac{f'(x)}{f(x)}dx = +\infty$ , if $\displaystyle\lim_{x\to\infty} f(x) = +\infty$ )

We assume that $\frac{\Delta S_k}{S_k}} \to 0$ otherwise the series would diverge trivially.

Let $L_k = \log S_k$ , then $\Delta L_k = \log S_{k+1} - \log S_k = \log \left( 1 +\frac{\Delta S_k}{S_k} \right) =\frac{\Delta S_k}{S_k} + O\left( \left(\frac{\Delta S_k}{S_k}\right)^2 \right)$

We observe that $\sum_{k=0}^n \Delta L_k = \log S_{n+1} - \log S_0$ diverges since $\log S_n \to \infty$ and now by the limit comparison test - both differences are always positive-, we get that $\sum_{k=0}^n\frac{\Delta S_k}{S_k}}$ diverges too.

Notation: $\Delta a_n = a_{n+1}-a_n$
March 21st 2011, 11:54 AM
Bruno J.

Very nice Paul!

Here's my solution:

It's well known that if $x_n>0$ and $\sum {x_n}$ converges , then $\prod (1+x_n)$ converges also. Let $A_n = \sum_{k=0}^n a_k$ . We have

$\prod_{k=1}^n (1+\frac{a_k}{A_{k-1}}) = \prod_{k=1}^n\frac{A_k}{A_{k-1}} = A_n/a_0$ . Now since $A_n \to \infty$ , we see that $\prod (1+\frac{a_k}{A_{k-1}})$ diverges, which implies that $\sum \frac{a_k}{A_{k-1}}$ diverges.
March 26th 2011, 10:57 PM
bkarpuz

Quote:

Originally Posted by Bruno J.

Very nice Paul!

Here's my solution:

It's well known that if $x_n>0$ and $\sum {x_n}$ converges , then $\prod (1+x_n)$ converges also. Let $A_n = \sum_{k=0}^n a_k$ . We have

$\prod_{k=1}^n (1+\frac{a_k}{A_{k-1}}) = \prod_{k=1}^n\frac{A_k}{A_{k-1}} = A_n/a_0$ . Now since $A_n \to \infty$ , we see that $\prod (1+\frac{a_k}{A_{k-1}})$ diverges, which implies that $\sum \frac{a_k}{A_{k-1}}$ diverges.

Your solution is same as what came to my mind when I saw the question.
For those who might want to see a reference for the relation between the sum and the product used by Bruno J, please see [Theorem 7.4.6, 1].
I quote the result for convenience.
Lemma. Let $\{x_{n}\}_{n\in\mathbb{N}}$ be a sequence of nonnegative numbers, then
$\sum_{n\in\mathbb{N}}x_{n}$ and $\prod_{n\in\mathbb{N}}(1+x_{n})$ converge or diverge together.

References
[1] L.S. Hahn and B. Epstein, Classical Complex Analysis, Jones and Bartlett Publishers, Inc., London, 1996.