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MATH256 Problem Sheet 1

1.1  Use repeated dierentiation to obtain all terms in the Taylor series for the

function

p(x) = 2x3 + 4x2 - 7x + 5

about the point x = 3.

1.2   (a)   (i)  Find a formula for the nth derivative of f (x), in the case where

f (x) =    1        lxl < 1.

(ii)  Hence obtain the full Taylor series for the function f (x) about the point x = 0.

(iii)  Can you see an easy way to confirm that your result is correct?

Hint: think about the types of innite series that you know how to sum.

(b)  Use the result of part (a) to deduce the full Taylor series about x = 0 for

(i)   g(x) =     1           (ii)   h(x) = arctan(x),        lxl < 1.

Hint:  what is the derivative of arctan(x)?

1.3  Use the Taylor series for f (c + h) and f (c - h) to obtain a centred difference formula for f\\ (c) with an O (h2 ) error. Make sure you retain enough terms to clearly establish the order of the error.

1.4 Suppose that the function g(t) has the convergent Taylor series expansion g(t) = j g(j)(0)  .

(a)  Show that

-h(h) g(t) dt = 2 j h2j+1 .

Show that

g(h) - 2g(0) + g(-h) h2

= g\\ (0) + O(h2 ).

Make sure you  include enough  terms in  the  Taylor series to clearly

establish the order of the error.

(c)  Using the results of parts (a) and (b), show that

-h(h) g(t) dt =  g(-h) + 4g(0) + g(h)] + O(h5 ).

1.5 Suppose the series

o

j=1

is such that

lAj+1l s klAj l,    for   j > N,    where   0 < k < 1. (a)  Find an upper bound for the magnitude of the tail

T =         Aj ,

j =N+1

in terms of lAN+1l.

What does this tell you about the error in the approximation

 

N

j=1