Converging Series: Solve Sum of (3^n + 4^n) / (3^n + 5^n)

[kmeado07]

Homework Statement

Show that the following series converges:

Homework Equations

Sum of (from n=1 to infinity) of [3^n + 4^n] / [3^n + 5^n]

The Attempt at a Solution

Some help on this question would be much appreciated as i really don't know how to start it. Thanks

 

[sutupidmath]

What i would do is break the series into two. then by observing that:


$$3^n+5^n>5^n=>\frac{1}{3^n+5^n}<\frac{1}{5^n}=>\frac{3^n}{3^n+5^n}<\left(\frac{3}{5}\right)^n$$


Also:

$$3^n+5^n>5^n=>... \frac{4^n}{3^n+5^n}<\left(\frac{4}{5}\right)^n$$