VISUALIZING MATHS & PHYSICS : DAMMIT ! DO TAYLORS SERIES HAVE A REASON TO EXIST IN MATH ?

Wednesday, 11 May 2022

DAMMIT ! DO TAYLORS SERIES HAVE A REASON TO EXIST IN MATH ?

Mathematicians feel difficult to work with e^x, sin(x), cos(x) etc for a variety of reasons which i will go into the patreon page of this article.

So that is when English Mathematician Brooke Taylor came up with a solution.

Brooke Taylor said e^x (or any other function) can be nearly fully duplicated/replicated/cloned using a summation of polynomial terms.

This artificial replication/cloning of functions (like e^x, cos(x),sin(x)) using a summation of polynomials is called as Taylor's series.

Just look at the beauty of this equation.

Look carefully.

e^x passes through 1 on the y-axis. So at x=0, y=1.

So when Brooke Taylor decided to recreate e^x using a summation of polynomials, does it surprise you that the first term of this summation is 1 ?

So,

The first term is 1
[I recommend seeing 1 as (x^0)/0!]
The second term is (x^1)/1!
Then the third term is (x^2)/2!
The fourth term should be (x^3)/3!

This summation goes on till (x^n)/n!

So what does this mean? It means that when you plot the graph of 1+ (x^2)/2! + (x^3)/3!......all the way up to (x^n)/n!, the graph starts looking similar to the graph of e^x.

Just trust me and do the following

Graphing on Desmos.com

e^x v/s 1+x+(x^2/2!) + (x^3)/3! +... (x^n)/n!

1) Go to Desmos.com online calculator.

2) Type and derive the graph of e^x

3) Type and derive another graph of 1+x+(x^2)/2!+(x^3)/3!+..........(x^6)/6!

4) Now compare the two graphs.

What you will see is that as the values of the n in the second equation increases.....the graph of [1+x+(x^2)/2!+(x^3)/3!+.......(x^n)/n!] starts resembling the graph of e^x.

So, how is this helpful?

Taylors series is cloning functions (like e^x, cos(x),sin(x) using polynomials as parts.) Mathematicians find it easier to deal with these parts than to deal with an entire function itself.

You might have heard of the old adage about eating an elephant.

e^x, cos(x), sin(x)....these are the elephants for mathematicians. Taylor's series is cloning the elephant using parts of something else (parts -----> are analogous to polynomials). And then eat/ deal with these parts separately.

Duplicating/cloning e^x using polynomials as parts means.... you can manipulate any of these parts of this polynomial of it as per your wish (customization).

Such a selective customization may not be possible with e^x or cos(x) or sin(x). Because whatever you do to these functions will reflect throughout the graph of e^x or sin(x) or cos(x).

Other important videos on Taylor's series.