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# FP2 Maclaurin's and Taylor's Series

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1. For this question, why do you not use Taylor's expansion with a=8? Is it only when a function is translated horizontally where you don't use a=0?

Also, why do you use x instead of e^x in the expansion? Would using e^x give you the same answer?

Thanks for any help!
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2. (Original post by PhyM23)
For this question, why do you not use Taylor's expansion with a=8? Is it only when a function is translated horizontally where you don't use a=0?

Also, why do you use x instead of e^x in the expansion? Would using e^x give you the same answer?

Thanks for any help!
The question wants the expansion in powers of and not powers of , which is where you'd use the Taylor expansion.

I don't understand your second question.
3. The series expansion of a function is taken from 0,0 if nothing else is stated as a baseline. Also it wants in powers of

Maclaurin series states that:

Hence is used in the expansion. However, would not give the same answer.
4. (Original post by Zacken)
The question wants the expansion in powers of and not powers of , which is where you'd use the Taylor expansion.

I don't understand your second question.

But why couldn't you use f(x) = f(a) + xf ' (a) ... with a=8?

I mean why do you use x(f'(0)) + ((x^2)/2)f''(0)? Why aren't the 'x's in this expansion 'e^x's?
5. (Original post by Cryptokyo)
The series expansion of a function is taken from 0,0 if nothing else is stated as a baseline. Also it wants in powers of

Maclaurin series states that:

Hence is used in the expansion. However, would not give the same answer.

But in my textbook it says 'Express tan(x+pi/4)as a series in ascending powers of x up to the term x^3' and they've used a=pi/4. Why don't they use a=0?
6. (Original post by PhyM23)

But why couldn't you use f(x) = f(a) + xf ' (a) ... with a=8?

I mean why do you use x(f'(0)) + ((x^2)/2)f''(0)? Why aren't the 'x's in this expansion 'e^x's?
What you've written down is incorrect.

f(x) = f(a) + xf ' a() + ...

The correct version is:

f(x) = f(a) + (x-a)f ' (a) + ...

Or:

f(x+a) = f(a) + x f' (a) + ....

But f(x+a) is not y, so is useless here.

Second of all, the question wants an expansion in powers of , not .
7. (Original post by PhyM23)

But in my textbook it says 'Express tan(x+pi/4)as a series in ascending powers of x up to the term x^3' and they've used a=pi/4. Why don't they use a=0?
You could use a=8 but it wants each power of x in its simplest form and

is not each term in its simplest form. If you expanded the brackets and then made each power of x into its simplest form it would give the correct answer. But this is much more involved.
8. (Original post by Zacken)
What you've written down is incorrect.

f(x) = f(a) + xf ' a() + ...

The correct version is:

f(x) = f(a) + (x-a)f ' (a) + ...

Or:

f(x+a) = f(a) + x f' (a) + ....

But f(x+a) is not y, so is useless here.

Second of all, the question wants an expansion in powers of , not .

Ah my apologies I did mean f(x+a). What you've said makes sense. Thanks for clearing this up.
9. (Original post by Cryptokyo)
You could use a=8 but it wants each power of x in its simplest form and

is not each term in its simplest form. If you expanded the brackets and then made each power of x into its simplest form it would give the correct answer. But this is much more involved.
Thank you for this; it's very helpful!
10. No problem!

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