DC Mathematica 2017

(𝑖) 2 2!

(𝑖) 3 3!

(𝑖) 4 4!

𝑖 1!

 𝑥 = 1 +

+

+

+

. . .

Notice that this becomes:

2 2!

𝑖 3 3!

4 4!

𝑖 5 5!

𝑖 1!

 𝑥 = 1 +

−

−

+

+

. . .

Grouping the terms gives:

2 2!

4 4!

6 6!

3 3!

5 5!

(1 −

+

−

. . . ) + 𝑖( −

+

. . . )

However, since the first parenthesis is the Taylor series for cos(x) and the second parenthesis is the Taylor series for sin(x), we can transform the equation into:

 𝑥 = () + sin()𝑖

Finally, evaluating at 𝜋 :

 𝜋 = −1

which is Euler’s identity.

Applications of Taylor Series

Apart from the unbelievable result obtained above, there are many other applications for this formula. For example, there is no way to evaluate the indefinite integral ∫ 𝑖
( 2 ) via conventional means. However, by substituting 2 =
into the equation for 𝑖
(
) and integrating each of the individual terms a reasonably accurate approximation can be found. On a similar note, if you ever find yourself having forgotten your calculator in an exam you can find a numerical value to an appropriate degree of accuracy for certain functions using a Taylor series (this is definitely not recommended). For those of you who are fortunate (or unfortunate) enough to have to solve differential equations on a regular basis, the Taylor series can be a weapon to add to your repertoire. For example:
’’ + 2
=  𝑥 . This differential equation is not at all trivial to solve, but by representing y as a Taylor series it’s possible to approximate a solution. Further uses include: evaluating infinite sums, generating functions in combinatorics, evaluating energy functions in physical systems at equilibrium, power flow analysis and many more.

I hope you found this article interesting and informative.

Challenge:

Bibliography:

Can you prove the result

i i =e - π/2

http://math.stackexchange.com/questions/218421/what- are-the-practical-applications-of-the-taylor-series

as given above?

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