converging sequence?

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.

Quote:

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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.

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

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

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