• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Taylor Series Evaluation

1. The problem statement, all variables and given/known data
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
2. Relevant equations
The Taylor Series Equation
3. The attempt at a solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the initial constant term ##f(a)## would be equal to 0, hence all I have to do is to evaluate ##f(a)##. If I were to extract ##f(x)## from the function, all I get is ##(-1)^n \sqrt n## and I'm unsure how to evaluate it from there
 

haruspex

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
30,837
4,453
all terms in the series except for the initial constant term f(a)f(a)f(a) would be equal to 0
Of f(8)(a), yes, but not of f(a).
Do you understand the notation?
 

Delta2

Homework Helper
Insights Author
Gold Member
2,090
573
Notice that it asks you to find the value of eighth derivative at ##x=4##. So you should take the derivative of that series expression ,8 times. To give you the hunch of how it goes, the first derivative is ##f'(x)=\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{(n-1)!}(x-4)^{n-1}##.

So first find the general expression for ##f^{(8)}(x)## and then evaluate it at ##x=4##. Give caution for what the base value of ##n## would be, as you see for the first derivative the base value of ##n## has become ##n=1##.
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,710
1. The problem statement, all variables and given/known data
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
2. Relevant equations
The Taylor Series Equation
3. The attempt at a solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the initial constant term ##f(a)## would be equal to 0, hence all I have to do is to evaluate ##f(a)##. If I were to extract ##f(x)## from the function, all I get is ##(-1)^n \sqrt n## and I'm unsure how to evaluate it from there
What is ##(\frac d {dx})^8 (x-4)^n## for ##n < 8,## ##n=8## and ##n > 8##? That will tell you what ##f^{(8)}(x)## looks like.
 

Delta2

Homework Helper
Insights Author
Gold Member
2,090
573
I see now what you mean at the OP, if we directly compare the given series, to the Taylor expansion of the function around point x=4, we can immediately conclude the formula for ##f^{(n)}(4)##.
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,710
1. The problem statement, all variables and given/known data
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
2. Relevant equations
The Taylor Series Equation
3. The attempt at a solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the initial constant term ##f(a)## would be equal to 0, hence all I have to do is to evaluate ##f(a)##. If I were to extract ##f(x)## from the function, all I get is ##(-1)^n \sqrt n## and I'm unsure how to evaluate it from there
The definition of a Taylor series is
$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n$$
where
$$f^{(n)}(a) \equiv \left. \frac{d^n f(x)}{dx^n} \right|_{x=a}$$
 

Want to reply to this thread?

"Taylor Series Evaluation" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top