I am trying to calculate the 11th derivative, , of a function using the Taylor Series centred at one.
I arrived to this stage where I form it into a geometry series format:
From here, I believe that the is essentially the r in a geometry series formula.
So I put that into a summation:
I cleaned it up a little bit and manipulated the algebra and arrived at:
Now, this looks like a Taylor Series format to me with at the back and as my co-efficient.
Since I got the co-efficient of the series, I thought I could just do this:
and then get:
However, my answer is wrong and when I do it on a calculator,
What should I do after this? I thought I can get the answer once I turn it into a format that looks like the original Taylor series format and find the answer from the co-efficient?
hmm...So the coefficient cannot have x in it.
I eventually reach my current stage from what I had in my previous post:
But I cannot figure out how to carry on from this to the given final answer, which is:
Any idea how I could arrive the final answer from my current stage?
Now the coefficient of the term of degree 11 in (1) is...
... so that the requested value should be...
Because in pure calculus I'm a little poor [...] it would be better if some young man or woman controls my computation...
I could understand the line (1). But I still couldn't figure out how to derive from (1) to line (2) to get the 11th term's coefficient. What was being substituted into the t? If I use back x=t+1, t=x-1, and with centred at 1, t would become 1-1=0?
thanks. it looks like a neat method. really wish to try it.
I substitute into and I get:
Then I bring it into the summation and expand it and get:
Since my coefficient cannot have X in it, and I have to make the whole thing look like only to match the Taylor Series, I need to somehow manipulate the algebra to look like that. But I couldn't seem to figure out how to do that because there is like so many (x-1) in . How do I continue from here?
Oh! I figure it out! Thanks! What a neat method! Thank you so much!
You can write it as if you are not comfortable with that notation.
For your problem, we have
Both and vanish after the 1st and 2nd derivative respectively.
So we only need to calculate the 11th derivative of .
If , then: