Question 977051
Test the series {{{1/(1*2*3) + 3/(2*3*4) + 5/(3*4*5) +""*""*""*""}}} for convergence or divergence.
<pre>
The nth term is {{{(2n-1)/(n(n+1)(n+2))}}}

The degree of the denominator is 2 more than the degree of the numerator,
so we use the limit comparison test to the p-series with p=2,
{{{sum(1/n^2,matrix(1,3,"","",n=1),matrix(1,4,"","","",infinity))}}}

which converges.

{{{matrix(2,2,

lim,    ((2n-1)/(n(n+1)(n+2)))/(1/n^2),
n=infinity,"")}}}

{{{matrix(2,2,

lim,    ((2n-1)/(n(n+1)(n+2)))*(n^2/1),
n=infinity,"")}}}

{{{matrix(2,2,

lim,    ((2n-1)/(cross(n)(n+1)(n+2)))*(n^cross(2)/1),
n=infinity,"")}}}

{{{matrix(2,2,

lim,    ((2n-1)/((n+1)(n+2)))*(n/1),
n=infinity,"")}}}

The degrees of numerator and denominator are the same,
the leading coefficient of the numerator is 2 and the
leading coefficient of the denominator is 1, so the
limit is 2/1 or 2.  Therefore the series converges.

Actually it converges to 3/4, but that takes more work to
discover that.

Edwin</pre>