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The University of Sydney

School of Mathematics and Statistics

Tutorial 10: The Euler Maclaurin Formula.

MATH3068 Analysis Semester 2, 2009

Web Page: http://www.maths.usyd.edu.au:8000/u/UG/SM/MATH3068/Lecturer: Donald Cartwright

Here is the general form of the Euler-Maclaurin Formula:

   n1

f (x) dx  =n

j=1

f ( j) −f (1) + f (n)

2  −

rj=1

B2j

(2 j)!

f (2j−1)(n) − f (2j−1)(1)

+  1

(2r)!

   n1

f (2r)(x) B̃2r(x) dx,

which is valid if  f (x) has a continuous 2r-th derivative. If  f (x) has a continuous 2r + 1-st derivative, thelast term can be re-written

1(2r)!

   n

1

f (2r)(x) B̃2r(x) dx  = − 1(2r + 1)!

   n

1

f (2r+1)(x) B̃2r+1(x) dx.

1.   Write down both forms of this formula for   r   = 1 and   r  = 2, evaluating the Bernoulli numbersinvolved.

2.   Apply the Euler-Maclaurin formula with r  = 1 and the second form of its final term to  f (x) = 1/x3.Our aim is to find an approximate value for the sum  ζ (3) of the infinite series

∞

k=1

1

k3.

3.   Use Stirling’s Formula to find

limn→∞

√ n

22n

2n

n

  and lim

n→∞

n

(n!)1

n

.

4.   We obtained Stirling’s Formula by applying to f (x) = lnx the simplest form of the Euler-Maclaurinformula:    n

1

f (x) dx  =n

j=1

f ( j)−f (1) + f (n)

2  −

   n1

f ′(x) B̃1(x) dx,

(which is the case   r   = 0 of the general formula above). Apply the case   r   = 1 of the generalEuler-Maclaurin Formula (with the second form of its final term) to  f (x) = lnx. What furtherinformation does this give us about  n!?

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