CS 270 Examples of Well-Known Series

1) Geometric Series:

=

converges for

<

2) Trigonometric Functions:

=

converges for all x

=

converges for all x

3) Exponential and Logarithmic Functions:

=

converges for -1 < x <= 1

=

converges for all x

4) Harmonic Series:

=

That is, this series diverges.

5) Calculations with Series:

a) Term-by-term differentiation of the series for sin(x) produces the series for cos(x).
Question: When is term-by-term differentiation legal?

b) An interesting manipulation of a series:

Start with the geometric series

=

and make the substitution

=

This gives

=

Question: Is this legal?

Then integrate both sides from 0 to 1. Assuming that it is OK to integrate term-
by-term, we get:

=

where

=

Thus we get:

=

Hence

=