Question
Asked Sep 7, 2019
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Taylor Expansion In class we used several results from Taylor expansion.
(a) Prove
In(1 -e) -
n
n-1
(b) Prove
e= cos 0 i sin 0.
Hint: Expand e, cos 0 and sin 0 using Taylor expansions and then compare the two
sides of the identity
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Taylor Expansion In class we used several results from Taylor expansion. (a) Prove In(1 -e) - n n-1 (b) Prove e= cos 0 i sin 0. Hint: Expand e, cos 0 and sin 0 using Taylor expansions and then compare the two sides of the identity

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Expert Answer

Step 1

(a)

Given statement is:

 

The Taylor expansion for ln (1 – x) is given by:

Plugging x = e in the above statement:

Therefore the given statement is true.

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In (1-e)- n-1 In (1-x) n=1 n In (1-e) n n-1 72 8

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Step 2

Given statement is:

 

Consider the left hand side of the statement:

The Taylor expansion for exponential function is given by:

Plugging x = iθ in the above equation and expanding the summation:

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e' cos i sin0 e 11 n! n 0 77 (i0)" n! n 0 (ie (10(10(i0) (ie) 4 (i0) eis -+ 1! 2! 3! 4! 5! 1 i0 02i0 e'6 + + 1 1! 2! 3! 4! 5!

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Step 3

Consider the right hand side of the given statement:

The Taylor expa...

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