# How do you use a Taylor series to prove Euler's formula?

Euler's Formula

Let us first review some useful power series.

Now, we are ready to prove Euler's Formula.

Proof

By rewriting as a power series,

by distributing the powers,

by separating the real part and the imaginary part,

by identifying the power series,

Hence, we have Euler's Formula

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To prove Euler's formula using a Taylor series, you can start with the exponential function ( e^x ) and use its Taylor series expansion. The Taylor series expansion for ( e^x ) is:

[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots ]

Now, substitute ( ix ) into this series, where ( i ) is the imaginary unit (( i^2 = -1 )). You get:

[ e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \cdots ]

Simplify each term:

[ e^{ix} = 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \cdots ]

Group the real and imaginary parts together:

[ e^{ix} = (1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots) + i(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots) ]

Now, recall the Maclaurin series expansion for ( \cos(x) ) and ( \sin(x) ):

[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots ] [ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots ]

Substitute these into the expression for ( e^{ix} ):

[ e^{ix} = \cos(x) + i\sin(x) ]

This is Euler's formula, which states that for any real number ( x ):

[ e^{ix} = \cos(x) + i\sin(x) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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