Mixing
On approximations using taylor series
0
$begingroup$
Suppose $f$ is diffrentiable many times and we want to expand $f(x)$ around say $x=0$ using taylor. Usually it is written
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + O(x^3) $$
or
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + f'''(0)/6 x^3 + O(x^4) $$
What is the difference between these two expansions? Does this $O(x^3)$ means that all terms higher than $x^3$ are zero? How can we decide whether to write the expansion in the first or the second way?
real-analysis calculus
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
$endgroup$
add a comment |
0
$begingroup$
Suppose $f$ is diffrentiable many times and we want to expand $f(x)$ around say $x=0$ using taylor. Usually it is written
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + O(x^3) $$
or
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + f'''(0)/6 x^3 + O(x^4) $$
What is the difference between these two expansions? Does this $O(x^3)$ means that all terms higher than $x^3$ are zero? How can we decide whether to write the expansion in the first or the second way?
real-analysis calculus
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
$endgroup$
$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34
add a comment |
0
0
0
$begingroup$
Suppose $f$ is diffrentiable many times and we want to expand $f(x)$ around say $x=0$ using taylor. Usually it is written
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + O(x^3) $$
or
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + f'''(0)/6 x^3 + O(x^4) $$
What is the difference between these two expansions? Does this $O(x^3)$ means that all terms higher than $x^3$ are zero? How can we decide whether to write the expansion in the first or the second way?
real-analysis calculus
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
$endgroup$
Suppose $f$ is diffrentiable many times and we want to expand $f(x)$ around say $x=0$ using taylor. Usually it is written
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + O(x^3) $$
or
$$ f(x) = f(0) + f'(0) x + f''(0)/2 x^2 + f'''(0)/6 x^3 + O(x^4) $$
What is the difference between these two expansions? Does this $O(x^3)$ means that all terms higher than $x^3$ are zero? How can we decide whether to write the expansion in the first or the second way?
real-analysis calculus
real-analysis calculus
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
asked Jan 14 at 3:10
Jimmy SabaterJimmy Sabater
2,632322
2,632322
$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34
add a comment |
$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34
$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34
add a comment |
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$begingroup$
Hi & welcome to MSE. The "O" being used is often called "big O" (to differentiate it from "little O" which is something different), with the Wikipedia article Big O notation doing a quite good job explaining it. Basically, it's a type of order of magnitude regarding the unknown value. In particular, though, it doesn't means the terms higher than $x^3$, or any other power of $x$, are all zero.
$endgroup$
– John Omielan
Jan 14 at 3:16
$begingroup$
I think I got it now. So, what it really means is that $f(x) - TAYLOR(f,degree2) = O(x^3)$ but $f(x) - TAYLOR(f,degree3) = O(x^4)$ so the second one is close to zero because $x^4$ approaches to $0$ faster than $x^3$ ?
$endgroup$
– Jimmy Sabater
Jan 14 at 3:24
$begingroup$
Note that $x^4$ approaches $0$ faster than $x^3$ only for $lvert x rvert lt 1$, but you can't always assume $x$ is within this bound as the valid region for a Taylor series expansion may include regions outside of this. As the Wikipedia article says, the big O notation "characterizes functions according to their growth rates". However, the upper bound constant for $Oleft(x^3right)$ and for $Oleft(x^4right)$ may be different, so you can't directly compare them to each other.
$endgroup$
– John Omielan
Jan 14 at 3:29
$begingroup$
As for your 2nd question, i.e., writing the expansion in the 1st or 2nd way you mention, or implicitly how many terms to include, this depends on several factors. For example, what is the purpose of the expansion (e.g., if it's to prove or check on a limit, you may only need a certain number), how much accuracy you want (note each Taylor series function expansion has its own rate of convergence, so this is something to check on), etc.? However, from my limited experience & knowledge, in real life situations, you will likely not need or want to go past the fourth or fifth term very often.
$endgroup$
– John Omielan
Jan 14 at 3:34