converging sequence?

**rcmango** · January 29th 2009, 01:15 PM

The sequence is defined by A1 = 1

ImageShack - Image Hosting :: 12426930zg2.jpg

for n >= 1 is decreasing.

and positive for all n.

need to show it converges.

this is a sequence, not a series.

So i'm confused as to what test i use?

thanks.

**Mush** · January 29th 2009, 02:23 PM

Originally Posted by rcmango

The sequence is defined by A1 = 1

ImageShack - Image Hosting :: 12426930zg2.jpg

for n >= 1 is decreasing.

and positive for all n.

need to show it converges.

this is a sequence, not a series.

So i'm confused as to what test i use?

thanks.

$\displaystyle \sum_{i = 2}^{\infty} \frac{1}{i(i+2)}$ converges, if:

$\int_2^{\infty} \frac{dx}{x(x+2)}$ has a finite value.

**Plato** · January 29th 2009, 02:46 PM

Think collapsing sums!
$\sum\limits_{k = 2}^\infty {\frac{1} {{k\left( {k + 2} \right)}}} = \frac{1} {2}\sum\limits_{k = 2}^\infty {\left[ {\frac{1} {k} - \frac{1} {{k + 2}}} \right]} \to \frac{5} {{12}}$

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