Is superlinear convergence always better than linear convergence?

[haya]

Homework Statement

[PLAIN]http://im2.gulfup.com/2011-04-01/1301686351321.gif

Homework Equations

superlinearly convergence

The Attempt at a Solution

[PLAIN]http://im2.gulfup.com/2011-04-01/1301686616101.gif

this is what i know about it, kindly help me

 

[grey_earl]

You should look up the definition of a linearly convergent sequence, and then see if one implies the other. P.ex., if a sequence is linearly convergent, then it is also superlinearly convergent (which would be b in your answers). Or, there are sequences which are linearly convergent, but not superlinearly convergent, vice versa, and some which are both (that would be c). And so on. Of course, only one is right.

 

[haya]

the definition of linear convergence is

http://disk04.arb-up.com/i/00030/qbvg9pj513m1_t.jpg

 

[grey_earl]

Ok, so assume you have a superlinearly converging sequence {p_n}. Then {c_n} is a zero sequence, so it converges to zero. So, especially (from some n on), it must be smaller than a constant M, right? What does that imply for your original sequence {p_n} in terms of linear convergence?