Convergence of (2^(n)+3^(n))/(4^(n)+5^(n)) using the Comparison Test

[Juggler123]

Decide (with justification) if the following series converges or diverges;

Sum(1,infinty) (2^(n)+3^(n))/(4^(n)+5^(n))

I've tried using the ratio test but I couldn't see that it was helping in any way, should I be using a different type of test for this problem? I really can't see where to start with this one.

 

[Dick]

Try and think of a clever comparison to which you can apply the ratio test. E.g. (2^n+3^n)/(4^n+5^n)<=(3^n+3^n)/(4^n+4^n). See, I substituted a larger numerator and a smaller denominator?

 

[Juggler123]

So if you apply the ratio test to the (3^(n)+3^(n))/(4^(n)+4^(n)) you find that this series converges as l<1 (l=3/4?). Is it then allowable to say that the original series converges as it is less than (3^(n)+3^(n))/(4^(n)+4^(n)) and therefore the limit must be lees than the limit of the above series and hence it must converge.

 

[Dick]

You tell me, ok? Look up the comparison test for series and make sure all the requirements are fulfilled. It's good practice.