Course Information.

In Part I we aim to understand the behaviour of infinite sequences of real numbers, meaning what happens to the terms as we go further and further on in the sequence. Do the terms all gradually get as close as we like to a limiting value (then the sequence is said to converge to that value) or not? We have to be precise and avoid some plausible but misleading ideas. We also have to understand the precise definition well enough to be able to use it when we calculate examples, though we will gradually build up a stock of general results the Algebra of Limits), general techniques and particular case so that we don't have think so hard when faced with the next example.

Part II is about infinite sums of real numbers: how we can make a sensible definition of that vague idea and then how we can calculate the value of an infinite sum - if it exists. Luckily we can add a finite sum of real numbers. So given an infinite sequence of numbers we can can, for each n>1, calculate the sum of the first n numbers from the sequence. In this way we get a sequence of partial sums. We can then use the results from Part I on this new sequence. If the sequence of partial sums converges we say the infinite sum converges.

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Question Sheets


Solution Sheets


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Lecture Notes

Notes Contents
Sequences
  • Part I Sequences
  • 2 Convergence . . . . . . . . . . . . . .. . . . . . . . . . . . . .16
    • 2.1 What is a Sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
    • 2.2 The Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
    • 2.3 The Definition of Convergence . . . . . . . . . . . . . . . . . . . . . . . . 19
    • 2.4 The Completeness Property for ℝ . . . . . . . . . . . . . . . . . . . . . . 25
    • 2.5 Some General Theorems about Convergence . . . . . . . . . . . . . . . . 29
    • 2.6 Exponentiation - a digression . . . . . . . . . . . . . . . . . . . . . . . . 31
  • 3 The Calculation of Limits . . . . . . . . . . . . . .. . . . . . . . . . . . . .34
    • 3.1 The Sandwich Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
    • 3.2 The Algebra of Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
  • 4 Some Special Sequences . . . . . . . . . . . . . .. . . . . . . . . . . . . .43
    • 4.1 Basic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
    • 4.2 New Sequences from Old . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
    • 4.3 Newton’s Method for Finding Roots of Equations - optional . . . . . . . 56
  • 5 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
    • 5.1 Sequences that Tend to Infinity . . . . . . . . . . . . . . . . . . . . . . . 59
  • 6 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
    • 6.1 The Subsequence Test for Non-Convergence . . . . . . . . . . . . . . . . . . . . . 64
    • 6.2 Cauchy Sequences and the Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . 68
    • 6.2.1 Proofs for the section - optional . . . . . . . . . . . . . . . . . . . . . . . 69
  • 7 L'Hôpital's Rule . . . . . . . . . . . . . .. . . . . . . . . . . 74
Series
  • Part II Series 
  • 8 Introduction to Series . . . . . . . . . . . . . .. . . . . . . . . . . . . .79
    • 8.1 The Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
  • 9 Series with Non-Negative Terms . . . . . . . . . . . . . . . . . . . . . . . . . .85
    • 9.1 The Basic Theory of Series with Non-Negative Terms . . . . . . . . . . . . .. .85
    • 9.2 The Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
  • 10 Series with Positive and Negative Terms . . . . . . . . . . . . . .. . . . . . . . 98
    • 10.1 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
    • 10.2 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
  • 11 Power Series . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .106
    • 11.1 The Radius of Convergence of a Power Series . . . . . . . . . . . . . . . . 107
    • 11.2 The n-th Root Test . . . . . . . . . . . . . . . . . . . . . . . . 107
  • 12 Further Results on Power Series - further reading . . . . . . . . . . . . . .114
    • 12.1 More General Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . 114
    • 12.2 Rearranging Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
    • 12.3 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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