Limit - Preliminary test for divergence or further testing

[KindofSlow]

Hello,
I'm using the preliminary test to determine if an infinite series is divergent or requires further testing.
Limit as n approaches infinity of 3^n/(2^n+3^n).
Looks to me like the denominator is always 2^n greater than the numerator so the limit is zero as n gets bigger and bigger.
The book says I am wrong - that the limit does not approach zero, but does not have an explanation.
Can anyone explain why this does not approach zero?
Thank you

 

[pka]

Hello,
I'm using the preliminary test to determine if an infinite series is divergent or requires further testing.
Limit as n approaches infinity of 3^n/(2^n+3^n).

When I taught this, I insisted that it be called the first test for divergent.
The sequence part of a series must be null, i.e. limit zero. Otherwise the series diverges,

That said, \(\displaystyle \dfrac{{{3^n}}}{{{2^n} + {3^n}}} = \dfrac{1}{{{{\left( {\frac{2}{3}} \right)}^n} + 1}} \to 1\)

Thus that series diverges.

 

[KindofSlow]

Thank you - much appreciated.