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

Alternating Series estimation theorem vs taylor remainder

1. Homework Statement

Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120

2. Homework Equations
Rn(x)=M(x-a)^(n+1)/(n+1)!

sin(x)=sum from 0 to ∞ of (-1)^n*x^(2n+1)/(2n+1)!

3. The Attempt at a Solution
For the alternating series test, I plugged in 1 for x in the sin x macluarin series and got 1/(2(n+1)+1)! <= 1/120 and got n=1
Then I used the Taylor's remainder theorem, got 1/(n+1)! <= 1/120 so n=4. why am I getting a different answer from Alternating series estimation theorem?
 
Last edited:

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,478
719
1. Homework Statement

Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120

2. Homework Equations
Rn(x)=M(x-a)^(n+1)/(n+1)!

sin(x)=sum from 0 to ∞ of (-1)^n*x^(2n+1)/(2n+1)!

3. The Attempt at a Solution
For the alternating series test, I plugged in 1 for x in the sin x macluarin series and got 1/(2(n+1)+1)! <= 1/120 and got n=1
Then I used the Taylor's remainder theorem, got 1/(n+1)! <= 1/120 so n=4. why am I getting a different answer from Alternating series estimation theorem?
Let's write out a few terms of the two series. The alternating series is$$
x -\frac{x^3} 3 +\frac {x^5} 5~...$$
The Taylor series is$$
0+x+0-\frac {x^3} 3 + 0 +\frac{x^5} 5~...$$Does that give you an idea?
 

Want to reply to this thread?

"Alternating Series estimation theorem vs taylor remainder" You must log in or register to reply here.

Related Threads for: Alternating Series estimation theorem vs taylor remainder

Replies
7
Views
4K
  • Posted
Replies
1
Views
2K
Replies
2
Views
899
  • Posted
Replies
2
Views
2K
  • Posted
Replies
3
Views
3K
  • Posted
Replies
2
Views
5K
  • Posted
Replies
3
Views
2K
  • Posted
Replies
3
Views
2K

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