Convergence and Continuity: Analysis Questions on Series and Limits

[pbialos]

I have two questions i hope you can help me with:

The first one is about series of positive terms like, for example $$\sum _{i=0}^{inf}(n^{1/n}-1)$$.
Can i say that as the square of this series of positive terms diverges by the "nth term test", the original series also diverges.
If i can't use this, could you please point me on the right direction in order to do determine the convergence of this series?.

My second question is about a limit by definition(lambda-epsilon proof). I have to find out if the following function is continuous and/or differentiable:
$$f(x,y) =\left\{\begin{array}{cc}\frac {x*y} {\sqrt x -\sqrt y},&\mbox{ if }x\neq y\\0, & \mbox{x=y}\end{array}\right$$

Everything i tried indicates that this function is continuous, but i can't prove that the limit when (x,y)->(0,0) of f is in fact 0.
I you want me to explain a little better what i have done just ask. I would really appreciate any help you could give me.
Many Thanks, Paul.

 

[Galileo]

A general hint for these kind of limit problems. When you want to prove such a limit exists and is zero, you usually need to find an upper-bound for that function which also tends to zero. This is basically the squeeze theorem.
There are probably other ways to do it, but playing around I found:

$$\left| \frac{xy}{\sqrt{x}-\sqrt{y}}\right| \leq \frac{|xy|}{\sqrt{x}+\sqrt{y}}= |x|\cdot \left| \frac{y}{\sqrt{x}+\sqrt{y}}\right| \leq |x|\sqrt{y}$$
The first inequality uses the triangle inequality on the denominator. The second is obvious.
The right side clearly goes to zero.

 

[pbialos]

thanks

Ok, I understand what you did, and i was thinking to do something very similar, but the inequalities didnt came up to my mind. I now have to find out if it is differentiable or not. Would i be on the right track if i use the theorem that says that if the partial derivatives are continuous in an "interval", then the function is also differentiable on that "interval"? I think i should use this instead of the differentiability limit, because i don't know which are the points to test for differentiability. Am i right?

Well i found out the partial derivatives and i realized that they are continuous everywhere except on (0,0) (I forgot to tell on the initial problem statement that the function is defined only for x and y postive or equal to 0), so i used the differentiability limit and i found that it was differentiable on (0,0).
Is my answer and PROCEDURE correct?

Do you have any hints for the first problem?

Thankfully, Paul.

 

[Galileo]

$$\left| \frac{xy}{\sqrt{x}-\sqrt{y}}\right| \leq \frac{|xy|}{\sqrt{x}+\sqrt{y}}$$

This is a mistake. The right side is smaller than the left side, since $\sqrt{x}+\sqrt{y} \geq |\sqrt{x}-\sqrt{y}|$

 

[pbialos]

checking

OOOUch, i didnt check your work correctly the first time, but now i see it was wrong. So then how could i prove this limit?
Assuming i could prove the continuity limit, i am pretty sure i could prove the differntiability also.
Grateful, Paul.

 

[iNCREDiBLE]

OOOUch, i didnt check your work correctly the first time, but now i see it was wrong. So then how could i prove this limit?
Assuming i could prove the continuity limit, i am pretty sure i could prove the differntiability also.
Grateful, Paul.

Polar Coordinates.

 

[Galileo]

Actually, it's pretty easy to see the limit doesn't exist (so no need to check for differentiability).

Note that the line y=x (where x>0) doesn't lie in the domain of f. The denominator can gets arbitrality small in any neighbourhood around (0,0). Therefore, if you choose an $\epsilon$-neighbourhood around (0,0), no matter how small, you can approach the line y=x from two sides inside that neighbourhood. Take a circle of radius $r<\epsilon$ and approach y=x from two sides. The product xy is nonzero near that line, but $\sqrt{x}-\sqrt{y}$ approaches $0^+$ and $0^-$ respectively.

About the sum, you use 'i' as your index, but have 'n' in the expression, did you mean:

$$\sum_{n=0}^{\infty}(n^{1/n}-1)$$

I`m not familiar with the 'n'th-term test'.

 

[pbialos]

ok

ok, so problem solved for the second one.
About the first one, it is as you said:
$$\sum_{n=0}^{\infty}(n^{1/n}-1)$$

And what i called the "n-th term root" is just the propertie that says that if a series $$\sum_{n=0}^{\infty}(A_n)$$ converges, then $$\lim_{n->\infty}A_n=0$$

I am sorry, but i didnt know how this property was called in english.

I found another way to solve this exercise but again i am not sure if what i did is correct:
I divided $$n^{1/n}-1$$ by $$n^{1/n}$$ and took the limit of this when n tends to infinite. It gave me 0, so i concluded that the series:
$$\sum_{n=0}^{\infty}(n^{1/n}-1)$$ and $$\sum_{n=0}^{\infty}(n^{1/n})$$ behave the same way(i think it is because the limit comparison test right??).
As the second series diverges, then the original series does also diverges.

I would really appreciate if you could tell me if any of the two ways i tried are correct.
I have my doubts about the second one specially, because i don't remember if i could use the limit comparison test when the limit A_n/B_n is equal to 0 with n tending to infinite.

Thank you, Paul.