Sequence And Series

Question 2

Write down the formula for nth term of an Arithmetic Sequence

Answer 2

Un = ( a + (n-1) d)

Question 3

Write down the formula for the nth term of a Geometric Sequence

Answer 3

Un = a r^(n-1)

Question 4

In iteration: What is the condition to show that re-arranged function g(x), will converge to its roots?

Answer 4

│g’(xo)│ < 1, where xo is estimated root

Question 5

In iteration: What converges faster: 1st order or 2nd order rearrangement?

Answer 5

2nd order

Question 6

In iteration: How do you know if an arrangement of k(x) is 1st order ?

Answer 6

│k’(xo)│ <1 , where xo is estimated rootand │k’(α)│ ≠ 0, where α is actual root

Question 7

How do you know if an arrangement of h(x) is 2nd order ?

Answer 7

│h’(xo)│ <1 , where xo is estimated rootand │h’(α)│ = 0, where α is actual root

Question 8

How do you get the fixed point (limit) of a non linear function?

Answer 8

Since fixed point means previous values are the same as following values: change the variable to x and solve.

Question 9

If asked for an approximate root of a function what is the graphical method for doing so?

Answer 9

Re-arrange fn and then split into 2 fns. Plot both fns and point of intersection is approx root.ex. Find an approx soln for: x^(3)-2x+3 = 0 Re-arrange: x^(3)=2x-3Split into 2 fns: y=X^(3) and y=2x-3Plot: pt on intersection is approx root.

Question 10

Once approx root is known what are steps to calculate exact root?

Answer 10

1. Re-arrange fn for x=2. Do TEST for suitable arrangement: |f’(xo)|<13. If suitable change re-arrangement to ITERATIVE form.3. Sub approx root, xo, and carry out iterative process.

Question 11

What is the condition for a geometric series to have a sum to infinity?

Answer 11

|r| <1

Question 12

Re-write (2 - x)^(-1) so you can use binomial theorem to calculate the terms of the geometric series.

Answer 12

1/2[1- (x/2)]^(-1) |x/2| < 1