# Taylor Series Remainder

#### hadroneater

1. The problem statement, all variables and given/known data
What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy?

2. Relevant equations
taylor series....to complicated to type out here

remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)
where a = 0 in this case
and
M >= |f^(n+1)(t)|

3. The attempt at a solution
I don't really get this question at all. I know that |R(0.25)| = 0.00001 <= M/(n+1)! * |x - a|^(n+1)
But how do I get M when |f^(n+1)(t)| is unknown? I don't even know what |f^(n+1)(t)| means!

Related Calculus and Beyond Homework News on Phys.org

#### clamtrox

3. The attempt at a solution
I don't really get this question at all. I know that |R(0.25)| = 0.00001 <= M/(n+1)! * |x - a|^(n+1)
But how do I get M when |f^(n+1)(t)| is unknown? I don't even know what |f^(n+1)(t)| means!
f(n+1)(t) is the n+1:th derivative of f(t). So is your plan to find the lowest upper bound for Mn? It might be easier (and more likely to be correct too) if you just calculated enough terms from the series until you have the desired accuracy.

#### hadroneater

But we aren't marked on that...it has to be through the remainder method.

Anyways, I know what f^(n+1)(t). I just don't know what to plug in for t. And after that, doesn't it just become a plug-and-check game for n until I get less than 0.00001?

#### HallsofIvy

Science Advisor
All derivatives of cosine are $\pm cosine$ or $\pm sine$. What is the largest possible value of a sine or cosine?

### 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