Why does ΣKQ^n become ΔQΣKnQ^(n-1)?

[foo9008]

Homework Statement

in the second picture (refer to the circled part) , i can understand the ΣKQ^n , but i don't understand the second one , why it will become ΔQΣKnQ^(n-1) ?

Homework Equations

The Attempt at a Solution

 

[BvU]

Can't read the exponents. Did you understand 5.13 ?

 

[foo9008]

Can't read the exponents. Did you understand 5.13 ?

not really

 

[BvU]

It all hinges on $$\Delta Q << Q \ \ \Rightarrow \ \ (Q + \Delta Q)^2 = Q^2 + 2 Q\, \Delta Q + (\Delta Q)^2\approx Q^2 + 2 Q \Delta Q$$ which is satisfied (maybe not in the first iteration, but later on it is) -- $\Delta Q$ gets smaller and smaller if you do things right.
(I filled in n = 2 for simplicity)

You could also see this as a differentiation ( ${dh\over dQ} = 2KQ {\rm \ \ or \ \ } \Delta h = 2 Q \Delta Q$ ) and then the method is basically the Newton method

 

[foo9008]

It all hinges on $$\Delta Q << Q \ \ \Rightarrow \ \ (Q + \Delta Q)^2 = Q^2 + 2 Q\, \Delta Q + (\Delta Q)^2\approx Q^2 + 2 Q \Delta Q$$ which is satisfied (maybe not in the first iteration, but later on it is) -- $\Delta Q$ gets smaller and smaller if you do things right.
(I filled in n = 2 for simplicity)

You could also see this as a differentiation ( ${dh\over dQ} = 2KQ {\rm \ \ or \ \ } \Delta h = 2 Q \Delta Q$ ) and then the method is basically the Newton method

removed

 

[BvU]

??