Convergence

**math sucks** · November 25th 2007, 09:16 AM

Let $\{a_n\}$ be a strictly increasing sequence of positive integers.
Can the series
$\sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots$
ever converge?

**topsquark** · November 25th 2007, 09:36 AM

Originally Posted by math sucks

Let $\{a_n\}$ be a strictly increasing sequence of positive integers.
Can the series
$\sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots$
ever converge?

At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
$\sum_{n = 1}^{\infty}\frac{1}{n}$
which is divergent.

-Dan

**math sucks** · November 25th 2007, 12:49 PM

Originally Posted by topsquark

At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
$\sum_{n = 1}^{\infty}\frac{1}{n}$
which is divergent.

-Dan

Can you prove your statement?

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