代写MTH1030 assignments

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代写MTH1030 assignments
MTH1030: Assignment 3, 2016

Ramanujan

 

The Rules of the Game

... are the same as those for the rst two assignments. This assignment is worth a total of

100 marks.

1 Convergence at the speed of light [50 Marks, each

subquestion is worth 10 marks]

The Indian mathematician Srinivasa Ramanujan1 discovered the formula

1



=

X1

n=0

p

8(4n)!(1103 + 26390n)

9801(n!)43964n :

Note that the series starts with n = 0. This means that in the following it makes sense to

talk about the 0th term and the 0th partial sum of this series.

This series converges at an amazing speed and was used in 1985 to compute the rst

17,526,100 digits of , which was the world record at the time.

To prove this identity is quite tricky, so let's do some things with it that are within our

reach.

a) Use the 0th term of Ramanujan's series to approximate . How many correct digits do

you get (counting the 3 at the start as the rst digit)? Do the same using the 0th and the

1st term of the series.2

1This is mathematician that the recent movie \The man who knew innity" is all about.

2To coax Mathematica into displaying the rst 40 digits of some number after the decimal point use the

command N[number, 40]. For example, to display 40 digits of  use N[Pi, 40].

1

b) Using the ratio test, verify that the series is convergent.

c) Let an denote the nth term of this series, starting with 0th term a0. It turns out that

an+1 < Lan

for n  1, where L is the limit that you calculated under b). Show that this implies that

an < Ln