Convergence of a Series with Square Roots and Cubic Terms

[danielc]

Homework Statement

Investigate convergence of \sum\frac{k+k^{1/2}}{k^{3}-4k+3}

The Attempt at a Solution

I am trying to use comparison test to investigate the convergence , but I am having trouble of finding a correct term to be compared to. please help

Homework Statement

The Attempt at a Solution

 

[Dick]

Based on it's behavior at large k, do you think it converges or diverges and why? Your first step in finding a comparison is to figure out if you want to compare with something larger or smaller.

 

[danielc]

Base on the behavior of k , I think the series will converges , so I would have to compare it with something larger (called it b) , thus , if b converges than the original series converges. However, I am having hard time figuring out what converging b I should choose which won't violate the property of original series. Could you give me some hint?

 

[Dick]

Sure. If you have a/b and you want to create something larger, then you want to increase the numerator and decrease the denominator. I'll give you a big hint. How about (k^3)/2 for the denominator? Remember you don't have to worry about small values of k. The comparison only has to hold for large k. Now the numerator? Keep it simple.

 

[danielc]

If I make numerator to be k+k=2k instead of k+k^1/2 , does that work?
but is it okay to ignore the -4k term in the original denominator ?

 

[Dick]

It works great. But we aren't really ignoring the -4k+3 in the denominator. We are just saying that for k large enough, that k^3-4k+3>(k^3)/2. I think in this case k>2. But it's not terribly important exactly where. You can always use (k^3)/100 if you want. The proof still works.

 

[danielc]

thank you very much , I think I got the idea now .