Malaurin series problem

apkb

just wondering what i can do with this ?? i know how to use the maclaurin series but i have never seen a question in this form before, its the = sign that i dont know what to do

Determine the Maclaurin series of each of the following functions:
cosh x =(e^x + e^−x)/2

any help would be appreciated

Clinker

The idea is that you should work out a MacLaurin Series for each side of the equation, by finding the necessary derivatives, etc. If the equation is true, the two series should be the same.

apkb

Clinker wrote: »

The idea is that you should work out a MacLaurin Series for each side of the equation, by finding the necessary derivatives, etc. If the equation is true, the two series should be the same.

would the maclaurin series of the overall function = 0 then ??

Fremen

Is that one subsection in a question? If so, perhaps "each of the following" refers to each function in the overall assignment.

I would read the equality as
"Cosh x (which is defined as) (e^x + e^−x)/2".

It's pointless to work out the macLaurin series for both sides, because by definition it's going to be the same for both sides.

apkb

no theres another function in the problem part (ii) but i am able to do that 1. sorry i just copied the question in.

i think i should just write out the function by taking both maclaurin series away from eachother like

=> a - a + b - b + c - c ...... + n - n .
( i know this is not accurate representation)

and just show that they will cancel up until the nth term

Clinker

Fremen wrote: »

Is that one subsection in a question? If so, perhaps "each of the following" refers to each function in the overall assignment.

I would read the equality as
"Cosh x (which is defined as) (e^x + e^−x)/2".

It's pointless to work out the macLaurin series for both sides, because by definition it's going to be the same for both sides.

I'm just trying to understand the psychology of the question here: justifying the given equation by getting the student to show that each side gives rise to the same Maclaurin Series. Even if it's presented as the definition, one might want to show that definition makes sense. You might think that's pointless, but some lecturers might feel it has a didactic purpose (I haven't ever set it as a problem myself!).

MathsManiac

I'm certain Fremen's interpretation is correct here. The original question was clearly only a part of a question, as the OP has confirmed. That is:

Determine the Maclaurin series of each of the following functions:
(i) cosh x =(e^x + e^−x)/2
(ii) ...something else...

So, in the context of reading part (i) only, this means, as Fremen has said:
Determine the Maclaurin series of the function cosh x, which is defined as (e^x + e^−x)/2

You're being given the definition of cosh(x) because you may not have encountered it before. You're being told the definition so that you can go and differentiate it.

hivizman

If you differentiate the right hand side (e^x + e^-x)/2, you get (e^x - e^-x)/2, which is the definition of sinh x. Differentiate (e^x - e^-x)/2 and you get back to (e^x + e^-x)/2, that is, cosh x. A simple substitution of x = 0 into these expressions shows that sinh 0 = 0 and cosh 0 = 1. This gives you enough information to work out the Maclaurin series for cosh x directly.

Edit: having just seen the previous posts, I think that the question setter doesn't want people to determine a Maclaurin series for cosh x separately from (e^x + e^-x)/2 but rather to go through something like the reasoning above to differentiate cosh x and to recognise that the even derivatives are all cosh x and the odd derivatives are all sinh x. Hence the Maclaurin series will consist only of even powers of x.

jamesmcgrane

I'm pretty sure that you are just supposed to find the maclauren series for cosh x. the reason there is an equals is because that is how cosh is defined. The examiner obviously wants you to express the maclauren series in terms of exponentials rather than in terms of sinh's.

apkb wrote: »

just wondering what i can do with this ?? i know how to use the maclaurin series but i have never seen a question in this form before, its the = sign that i dont know what to do

Determine the Maclaurin series of each of the following functions:
cosh x =(e^x + e^−x)/2

any help would be appreciated