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The University of Sydney

School of Mathematics and Statistics

Solutions to Assignment 1. Due date: Monday, April 23

MATH3068 Analysis Semester 1, 2007

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

1. Letxn= 1

n+1+ 1

n+2+ + 1

2n.

(a) Show that the sequence (xn) is monotonic and bounded and hence convergent. [Hints:a) Look atxn+1 xn, and b) how many terms are there in the sum making up xn? Whatis the greatest of these terms?]

Solution: For alln 1,

xn+1 xn= 1

n+ 2+

1

n + 3+ +

1

2n + 2

1n+ 1

+ 1

n + 2+ +

1

2n

=

1

2n + 1+

1

2n + 2

1

n + 1

= 1

2n + 1

1

2n + 2

>0.

Hence (xn) is monotonic increasing. Next,

xn= 1

n + 1+

1

n + 2+ +

1

2n