Mixing
How can I determine the end behavior of a polynomial based on Taylor series?
$begingroup$
I know this represents the cos(x) function, but what does it mean that the summation starts with 1 and ends with 21?
This question is for my review sheet for an exam I have. Also, if I were to write the last three terms, does that mean they would be $x^6/6!, x^8/8!$, and $x^{10}/10!$ ?
algebra-precalculus taylor-expansion
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
$endgroup$
add a comment |
$begingroup$
I know this represents the cos(x) function, but what does it mean that the summation starts with 1 and ends with 21?
This question is for my review sheet for an exam I have. Also, if I were to write the last three terms, does that mean they would be $x^6/6!, x^8/8!$, and $x^{10}/10!$ ?
algebra-precalculus taylor-expansion
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
$endgroup$
$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
1
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22
add a comment |
$begingroup$
I know this represents the cos(x) function, but what does it mean that the summation starts with 1 and ends with 21?
This question is for my review sheet for an exam I have. Also, if I were to write the last three terms, does that mean they would be $x^6/6!, x^8/8!$, and $x^{10}/10!$ ?
algebra-precalculus taylor-expansion
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
$endgroup$
I know this represents the cos(x) function, but what does it mean that the summation starts with 1 and ends with 21?
This question is for my review sheet for an exam I have. Also, if I were to write the last three terms, does that mean they would be $x^6/6!, x^8/8!$, and $x^{10}/10!$ ?
algebra-precalculus taylor-expansion
algebra-precalculus taylor-expansion
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
edited Jan 24 at 18:31
J. W. Tanner
3,2801320
3,2801320
asked Jan 24 at 10:10
user8358234user8358234
342110
asked Jan 24 at 10:10
user8358234user8358234
342110
asked Jan 24 at 10:10
user8358234user8358234
342110
342110
$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
1
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22
add a comment |
$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
1
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22
$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
1
1
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $pminfty$ as $xtoinfty$ or $xto -infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
$endgroup$
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $pminfty$ as $xtoinfty$ or $xto -infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
$endgroup$
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
add a comment |
$begingroup$
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $pminfty$ as $xtoinfty$ or $xto -infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
$endgroup$
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
add a comment |
$begingroup$
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $pminfty$ as $xtoinfty$ or $xto -infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
$endgroup$
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $pminfty$ as $xtoinfty$ or $xto -infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
answered Jan 24 at 10:18
jmerryjmerry
14.4k1629
14.4k1629
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
add a comment |
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
so if it is even then sign is positive and if it is odd then sign is negative?
$endgroup$
– user8358234
Jan 24 at 10:29
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
Since you haven't specified what "it" is, out of several possible parameters, I can't answer your question. Or, for that matter, what "sign" refers to.
$endgroup$
– jmerry
Jan 24 at 10:46
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
$begingroup$
how does the summation from 1 to 21 affect the way the polynomial behaves? how it rises and falls? and how would i write the last three terms of this polynomial?
$endgroup$
– user8358234
Jan 24 at 11:07
add a comment |
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$begingroup$
Is it implied that we are looking at $x to infty$? In that case, you can just look at the last (=highest) term of the polynomial.
$endgroup$
– Matti P.
Jan 24 at 10:16
1
$begingroup$
The last term is $-x^{42}/42!$, not $x^{10}/10!$
$endgroup$
– TonyK
Jan 24 at 10:22