Infinite Series: Determine Convergence/Divergence

[XJellieBX]

Homework Statement

$$\sum\frac{7^{k}}{5^{k}+6^{k}}$$
Determine if this infinite series (from k=0 to infinity) converges or diverges.


2. The attempt at a solution
I set a_k=$$\frac{7^{k}}{5^{k}+6^{k}}$$
then I took the Ln of both sides
ln a_k=ln$$\frac{7^{k}}{5^{k}+6^{k}}$$=ln7^k-ln(5^k+6^k)

I'm not sure if I did it right or where to go from here.

 

[lanedance]

hi XjellieBX
do you know how to test for divergence or convergence?

 

[XJellieBX]

we learned the root test, the ratio test, and the basic comparison test in class. but I'm not sure which one to use.

 

[ascapoccia]

no need to take the natural log; that is making your life too hard. have you tried to look at a comparison test with a special type of series (geometric, p-series, harminic, alternating, etc)?

 

[XJellieBX]

yes. i tried to compare it to the geometric series, but i was having some problems with the denominator

 

[lanedance]

i think the ratio test would work well here

 

[kof9595995]

comparison test is ok, hint: 5^k+6^k<2*6^k