Unraveling the Mysteries of Taylor's Theorem

[learningphysics]

The n's in the fraction were increasing too though?

Ah yes, you're right. I messed up... should be:

ln(x+\delta) = ln(x) + \Sigma_{n=1}^{n=\infty}(-1)^{n+1}\frac{(\delta)^n}{nx^n}

 

[Oblio]

Could you do a step by step to getting that from the equation? (please)

 

[learningphysics]

Could you do a step by step to getting that from the equation? (please)

Ok we start with this:

ln(x+\delta) = ln(x) + \frac{1}{x}\delta - \frac{1}{2x^2}\delta^2 + \frac{1}{3x^3}\delta^3 - \frac{1}{4x^4}\delta^4

instead of a series... let's just look at the sequence...

\frac{1}{x}\delta, -\frac{1}{2x^2}\delta^2, \frac{1}{3x^3}\delta^3, -\frac{1}{4x^4}\delta^4

first term (n=1) is \frac{1}{x}\delta
second term(n=2) is -\frac{1}{2x^2}\delta^2

In general what is the nth term?

 

[Oblio]

Ok we start with this:

ln(x+\delta) = ln(x) + \frac{1}{x}\delta - \frac{1}{2x^2}\delta^2 + \frac{1}{3x^3}\delta^3 - \frac{1}{4x^4}\delta^4

instead of a series... let's just look at the sequence...

\frac{1}{x}\delta, -\frac{1}{2x^2}\delta^2, \frac{1}{3x^3}\delta^3, -\frac{1}{4x^4}\delta^4

first term (n=1) is \frac{1}{x}\delta
second term(n=2) is -\frac{1}{2x^2}\delta^2

In general what is the nth term?

lol well I know that it would be your answer!
How come the -1 requires the ^{n+1} but the others are just ^{n}?

 

[learningphysics]

lol well I know that it would be your answer!
How come the -1 requires the ^{n+1} but the others are just ^{n}?

because the n = 1 term needs to have a plus sign...then the signs alternate... if I use n instead of n + 1... then the signs won't come out right... the n=1 term will come out with a - sign... ie: -(1/x)delta... and all the signs will be the opposite of what they need to be... test it out...

 

[Oblio]

Right, ok.
So, are n's basically placed where things count up?

 

[learningphysics]

Right, ok.
So, are n's basically placed where things count up?

Yeah.

 

[Oblio]

lol that's not so hard... I thought I had to make the n's count...