Quantitative Methods
Lecture 5
Sequences, series
and infinite sums
•BAKVM

Outline of the lecture



A sequence of real numbers



A sequence of real numbers



A sequence of real numbers



Arithmetic progression



Arithmetic progression: the sum of the first n elements



Geometric progression



Geometric progression: the sum of the first n elements



Geometric progression: the sum of the first n elements



The limit of a sequence:  the neighbourhood of a point
•The neighbourhood of a point is the set of all “nearby” points:
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•A refined neighbourhood of the point is a smaller set of all “nearby” points:
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•And we are about to consider smaller and smaller neighbourhoods…

The limit of a sequence



The limit of a sequence



The limit of a sequence



The limit of a sequence



The limit of a sequence



The limit of a sequence



Properties of the limit



Properties of the limit



The limit as a linear mapping



Theorem



Infinite series



Sequence of partial sums



The sum of a series



Divergent series



The sum of a geometric series