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Definition

A power series is a series of the form:

A power series may converge for some values of x and diverge for other.

The sum of the series is a function

f(x) = c₀ + c₁x + c₂x² + c₃x³ + ...

whose domain is the set of all x for which the series converges.

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General power series

If c_n= 1 then

which is convergent when -1 < x < 1 and is divergent when |x| > 1.

The general power series is (a power series centered at a or a power series about a.)

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Theorem

Given a power series, there are only three possible cases:

The series converges only when x = a.

The series converges for all x.

There is a positive number R such that the series converges if |x – a| < R and diverges if |x - a| > R.

We call R the radius of convergence. The interval of

convergences of a power series is the interval that consists of all values of x for which the series converges.

Convergence for |x–a| < R

a–R a a+R

divergence divergence

Definition of radius of convergence

The radius of convergence is defined based on three possible cases:

If the series converges only when x = a, the radius of convergence is zero.

If the series converges for all x, the radius of convergence is ’.

If there is a positive number R such that the series converges if |x – a| < R and diverges if |x - a| > R, the radius of convergence is R.

Example of convergent series

Series Radius of Interval of convergence convergence

Geometric series R = 1 (-1, 1)

Example 1 R = 0 {0}

Example 2 R = 1 [2, 4)

Example 3 R = ’ (-’, ’)

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Ratio Test

Exercise

Find the radius of convergence and interval of convergence of the series

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Exercise

Find the radius of convergence and interval of convergence of the series

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∴Š‡•‡”‹‡•…‘˜‡”‰‡•™Š‡͖βšβ͘Ǥ

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ƒ”‘‹…•‡”‹‡•Ǧㆋ˜‡”‰‡

׵ ൌ ͵ǡ ൌ ͳƒ† ‹–‡”˜ƒŽ‘ˆ…‘˜‡”‰‡…‡ሾʹǡ Ͷሻ

Find the radius of convergence and interval of convergence of the series

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Exercise

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Representations of functions as power series

If |x| < 1,

Express as the sum of a power series and find the radius of convergence.

Answer:

Then, -1 < x² < 1, 0 < x² < 1, -1 < x < 1.

The radius of convergence is 1.

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ݔ^ଶ௡Ǥ

Find a power series representation for

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Exercise

Find a power series representation for

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Exercise

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Theorem

If f has a power series representation (expansion) at a

where | x – a | < R.

Then its coefficients are given by the formula

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Taylor series

The Taylor series of the function f at a is

The special case (a = 0) is called the Maclaurin series

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ʹǨ ݔ^ଶ ൅ ݂ԢԢԢሺͲሻ

͵Ǩ ݔ^ଷ ൅ ڮ

Theorem

If f(x) = T_n(x) + R_n(x) where T_n is the n^th degree Taylor polynomial of f at a and

for |x - a| < R then f is equal to the sum of its Taylor series on the interval |x - a| < R.

௡Ž‹՜ஶܴ^௡ ݔ ൌ Ͳ

Determine the Maclaurin series of f(x) = e^x and its radius of convergence.

Exercise

Determine the Maclaurin series of f(x) = sin(x) and its radius of convergence.

Exercise

Estimate the error of Taylor polynomials

Suppose

The n^th degree Taylor polynomial is

The absolute remainder

| R_n(x) | = | f(x) – T_n(x) |

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ሺݔ െ ܽሻ^௜

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ሺݔሻ

Estimate the error of Taylor polynomials

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Ǣ c ‹•„‡–™‡‡a ƒ†x

”—‡ˆ‘”‘‡•’‡…‹ˆ‹…˜ƒŽ—‡‘ˆc ‘–Š‡‹–‡”˜ƒŽa ƒ†x

Estimate the value of cos(1) by a Taylor polynomial of degree 6 at a = 0.

How accurate is this estimation?

What is the maximum error possible in using the estimation?

Exercise