Analyzing Quadratic & Linear Convergence

[muso07]

Homework Statement

Which of the following converge quadratically and which converge linearly?
a) 1/n^2
b) 1/2^(2n)
c) 1/sqrt(n)
d) 1/e^n

Homework Equations

All I've got in my lecture notes is: The sequence converges with order a if there exist constants a and C and integer N such that |x_(n+1) -x*| <= C|x_n -x*|^a, and n>=N.

The Attempt at a Solution

For a), I got the terms 1, 1/4, 1/9, etc, and concluded that it must converge linearly since I couldn't find a constant C such that x_(n+1) <= C(x_n). But I'm not sure if it's right, and I don't know how to explain it if it is right.

For b), I also had that it converges linearly, for the same reasons as above. Again, don't know how to explain it if it's right.

c) Same

d) Same

My answers seem really wrong because I don't think all of them converge linearly but I can't seem to find any constants C such that it works for quad convergence. I'm so confused..

 

[jbunniii]

For (a), it's certainly true for all $n$ that

$$\left| \frac{1}{(n+1)^2} - 0 \right| \leq \left| \frac{1}{n^2} - 0 \right|$$

(i.e., $C = a = 1$) so this shows that the sequence converges (to zero) at least linearly.

If it were to converge quadratically, you would have to furnish a constant $C$ such that

$$\left| \frac{1}{(n+1)^2} \right| \leq C \left| \frac{1}{n^2} \right|^2$$

for sufficiently large $n$.

But this is equivalent to

$$n^4 \leq C (n+1)^2$$

so it's clear that no such $C$ exists, since the left hand side grows at a higher order than the right hand side. Therefore the sequence in (a) does NOT converge quadratically.

Try writing out in detail a similar argument for the other parts. If you can show me in detail what you have done for part (b) I'll try to help you find a suitable constant, if it exists, or to prove that one cannot exist.

 

[muso07]

Wow, your explanation made so much more sense than the one in the lecture notes. So they all converge linearly, since they are all decreasing sequences.

b) If it converges quadratically, then

[tex]\left|\frac{1}{2^{2(n+1)}}\right|\leq C\left|\frac{1}{2^{2n}}\right|^{2}[/tex]

[tex](2^{n})^{4}\leq C(2^{n+1})^{2}[/tex]

which won't work because the LHS grows at a higher order, like you said about the first one. So it converges linearly.

c) Quadratically:

[tex]
\left|\frac{1}{\sqrt{n+1}}\right|\leq C\left|\frac{1}{\sqrt{n}}\right|^{2}
[/tex]

[tex]
n \leq C\sqrt{n+1}
[/tex]

Again won't work because LHS grows at a higher order. So it converges linearly.

d) Quadratically:

[tex]
\left|\frac{1}{e^{n+1}}}\right|\leq C\left|\frac{1}{e^{n}}\right|^{2}
[/tex]

[tex]
(e^{n})^{2} \leq Ce^{n+1}
[/tex]

Same as previous.

Is that right? It seems a bit weird that my lecturer would give us four sequences that converge linearly...

 

[jbunniii]

Is that right? It seems a bit weird that my lecturer would give us four sequences that converge linearly...

I agree it's a bit weird, but I believe your answers are correct.

Check out this Wikipedia page to see some examples of sequences that converge quadratically:

http://en.wikipedia.org/wiki/Rate_of_convergence

You need to have more rapid decay to achieve quadratic convergence, for example

$$\left\{\frac{1}{2^{2^n}}\right\}$$

 

[muso07]

Cheers, that helped a lot. :)