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# 7.1 Some Examples.

In definition 5.1 , we defined a sequence in to be a function . Since we are identifying with a subset of , every sequence in is also a sequence in , and all of our results for complex sequences are applicable to real sequences.

7.1   Notation () I will say  consider the sequence " or  consider the sequence " to mean  consider the sequence such that for all ". The arrow is read  maps to".

7.2   Definition (Geometric sequence.) For each , the sequence

is called the geometric sequence with ratio .

I will often represent a sequence in by a polygonal line with vertices . The two figures below represent geometric sequences with ratios and respectively.

7.3   Definition (Geometric series.) If , then the sequence is called the geometric series with ratio .

Figure b shows the geometric series corresponding to the geometric sequences in figure a. If you examine the figures you should notice a remarkable similarity between the figure representing and the figure representing .

7.4   Entertainment. Describe the apparent similarity between the figure for and the figure for . Then prove that this similarity is really present for all .

7.5   Definition (Constant sequence.) For each , let , denote the constant sequence ; i.e., .

Next: 7.2 Convergence Up: 7. Complex Sequences Previous: 7. Complex Sequences   Index