How do I find out if a series is convergent or divergent?

Question

anonymous · Answer

$$\sum_{k=1}^{infinity} (3*(1/4)^{k}-2*(1/5)^{k})$$

anonymous · Answer

this one converges... -> http://en.wikipedia.org/wiki/Ratio_test

anonymous · Answer

Ok, that was useful. How do I find the sum of this series? Is there a formula I can use and how would it look like with the particular example?

anonymous · Answer

each term is a geometric progression
first term has common ratio of 1/4

second term has common ratio of 1/5

formula for sum to infinity of a geometric = a /(1-r)      a = ist term, r = common ratio

so in this case sum to infinity   = (3/4) / 1 - (1/4)    -  (2.5) / 1 - (1/5) =  1 - 1/2
= 1/2

anonymous · Answer

Awesome! Thank you, jimmyrep! Have a nice day! :)

anonymous · Answer

formula for n terms of this series is

[(3/4)*(1 - (1/4^n)]  /  3/4  -   [ (2/5) (1 - (1/5)^n] / (4/5)

 no probs