Further Sequences and Series

Example: Express ex as a power series. Step 1 Step 2 Step 3 Step 3 This process can be repeated indefinitely.

This is usually written; This is often referred to as Maclaurin’s expansion for ex. Let us consider the same technique used on the general function f(x).

Step 1 Step 2 Step 3 Step 3 This process can be repeated indefinitely.

This can be expressed more concisely as The region for which it is valid can be ascertained by considering

Use the Maclaurin series to find the first six terms of an expansion for (1 + x)½

Page 91 Exercise 4 Question 1

Let us explore a power series for a trig function. Repeating:

Composite Functions We know that for all k, This is NOT a power series in x.

Knowing: + Expanding and bringing together like terms……

+

Page 95 Exercise 6, Question 7 and 8 TJ Exercise 1 and 2

Iteration An iterative sequence is a sequence generated by a recurrence relation of the form xn+1 = F(xn). Each term of the sequence is known as an iterate, xn being the nth iterate. The term x0 is referred to as the starting value of the iteration. If, for some value of n, xn+1 = F(xn) then xn is called a fixed point in the sequence. A fixed point is often referred to as a convergent. When solving the equation f(x) = 0 graphically, we would normally draw the graph of y = f(x) and focus on where the curve cuts the x axis. If y = x2 – 3x + 2, then the graph indicates that the solutions are x = 1 and x = 2.

However, a simple rearrangement to shows us that it would be equally valid to draw the graphs of and where they intersect. The spreadsheet shows that with a starting value of 0.5, the sequence converges on the solution x = 1. Two other obvious rearrangements are: Converges on x = 1 Converges on x = 2

Thus if the equation f(x) = 0 can be rearranged to the form x = F(x), then studying the iterative sequence defined by xn+1 = F(xn) for its convergents may lead to solutions of the equation. Different rearrangements may lead to different solutions. Page 99 Exercise 8.

Iterative Process y x Staircase Diagram

Cobweb Diagram

Iterative Process Taylor Series Note the Maclauren series is obtained by letting x = 0 and h = x The main problem faced by us is whether or not the sequence converges to a solution, and if it does, how quickly.

The sequence will converge if: The smaller this value the faster the convergence. The proof of this is on page 100.

Since f(x) changes sign a root lies in the region

(i) Since we know a ≠ 0 then it follows F’(a) ≠ 0 (i) Since we know a ≠ 0 then it follows F’(a) ≠ 0. Thus the process is first order.

We need three consecutive iterates.

Thus x = 0.653 is a root correct to 3 decimal places.

Higher-order Processes An iterative process where this is the case is defined as a second order iterative process. The higher order the process, the faster the rate of convergence. However, the idea that we should construct higher-order processes is countered by the fact that each iterate then becomes quite complicated to compute. Page 102 Exercise 9. Questions 1, 2 and 4(VERY HARD)