9.1 Power Series

Quick Review

What you’ll learn about Geometric Series Representing Functions by Series Differentiation and Integration Identifying a Series Essential Question How can we use power series in understanding the physical universe and how can this be used to represent functions.

Infinite Series An infinite series is an expression of the form The number a 1, a 2,... are the terms of the series; a n is the nth term. Example Identifying a Divergent Series 1.Does the series 2 – – – converge? If the infinite series has a sum it has to be the limit of its partial sums, Since the sequence has no limit, the series has no sum.

Example Identifying a Convergent Series The sequence of partial sums, written in decimal form is: This sequence has a limit of

Infinite Series If the infinite series Geometric Series The geometric series

Example Analyzing Geometric Series 3.Tell whether each series converges or diverges. If it converges, give its sum.

Power Series An expression of the form is a power series centered at x = 0. An expression of the form is a power series centered at x = a.

Example Finding a Power Series by Differentiation

Term-by-Term Differentiation obtained by differentiating the series for f term by term, converges for

Example Finding a Power Series by Integration

Term-by-Term Integration obtained by integrating the series for f term by term, converges for

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