Convergence or Divergence: What Does the Limit of the Series Reveal?

[wheeler90]

1. $$\sum(\sqrt{k^{2}+1}-\sqrt{k^{2}})$$ from K=0 to K=$$\infty$$




2. Hi all. I need some help here. I have to use a test to determine whether the sum series diverges or converges



3. I thought it was the divergence test because I thought that the limit of the sum didn't approach zero, but I think I'm wrong. I need some help here. Thanks

 

[Dick]

The limit of the kth term does approach 0. But that doesn't mean it converges. Try multiplying by (sqrt(k^2+1)+sqrt(k^2))/(sqrt(k^2+1)+sqrt(k^2)) and simplify the algebra. Then give me your opinion about convergence.

 

[Office_Shredder]

If you multiply

$$(\sqrt{k^2+1} - \sqrt{k^2})(\sqrt{k^2+1} + \sqrt{k^2}) = 1$$ So what you actually have is the series

$$\sum \frac{1}{\sqrt{k^2+1} + \sqrt{k^2}})$$

Does that help?

 

[wheeler90]

Thank you both. That helped a lot. I used the comparison test and the p-series test and it does converge.

The real problem was similar in that instead of $$\sqrt{k^{2}+1}$$ - $$\sqrt{k^{2}}$$ it was $$\sqrt{k^{5}+10}$$ - $$\sqrt{k^{5}}$$. I just didn't want it to feel like cheating. Thanks again.