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Operations On Power Series: 1+x+x +X +... - 1 X 1

The document discusses operations that can be performed on power series. It introduces common power series like the Taylor series for e^x and (1-x)^-1. It presents two theorems - Theorem A discusses term-by-term differentiation of a power series, and Theorem B states that arithmetic operations can be performed term-by-term on convergent power series. Examples demonstrate finding power series representations for functions like sinh(x), computing sums of power series, and using the theorems to derive and manipulate other power series.

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Saddie Soula
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53 views8 pages

Operations On Power Series: 1+x+x +X +... - 1 X 1

The document discusses operations that can be performed on power series. It introduces common power series like the Taylor series for e^x and (1-x)^-1. It presents two theorems - Theorem A discusses term-by-term differentiation of a power series, and Theorem B states that arithmetic operations can be performed term-by-term on convergent power series. Examples demonstrate finding power series representations for functions like sinh(x), computing sums of power series, and using the theorems to derive and manipulate other power series.

Uploaded by

Saddie Soula
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Operations on Power Series

= 1+x+x2+x3+... -1<x<1

= 1+2x+3x2+4x3+... -1<x<1

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Operations on Power Series

Think of a power series as a polynomial with infinitely many terms.

Theorem A

Let on the interval, I.

If x is interior to I, then

1)

2)

EX 1 We know

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EX 2 Show S'(x)=S(x) for .

You must first demonstrate convergence, then solve S'(x)=S(x).


Notice S(0) = 1.

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EX 3 Find the power series for .

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Theorem B

If and with both series converging

for |x|<r, we can perform arithmetic operations and the resulting


series will converge for |x|<r. (If b00, the result holds for division,
but we can guarantee its validity only for |x| sufficiently small.)

EX 4 Find a power series for f(x) = sinh(x).

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EX 5 Find the power series for .

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EX 6 Find these sums.

a)

b) cos x+ cos2x+cos3x+cos4x+...

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