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ERJAEE, G. (20)

Lecture 27

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

27This

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MATH 2B - Lecture 27 - Representations of Functions as Power Series

We first recall the geometric series

●

Expanding it, we obtain

●

Therefore, we can shift the index n by 1, and obtain a new form for geometric series

●

Now, we consider a simple power series

●

● We can treat this power series as a geometric series by letting

Similarly, we have

●

● Therefore, we can think this power series as a function. But this only hold on a

certain interval

● So, a power series is equal to a function only on its interval of convergence

● Now, we try to represent some functions by using power series, and we also

need to find the interval of convergence

Example:

● Represent

by power series

Solution:

● We always need the geometric series

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