Maclaurin Series and the general term

[maccaman]

using the sin and cos Maclaurin series, validate each of them using at least 3 values for x and determine how many terms are needed to provide reasonable accuracy.

Find the General Term (Tn where n = 1, 2, 3, ...) for each expression and show that each correctly generates the terms of the series. Then using the specific and General Terms, determine expressions for the derivative of sin x and the integral of cos x.



Now I get the part where i must use 3 values for x and how many terms, but what has really got me stumped is the 2nd paragraph. Any help would be greatly appreciated. Thanks

 

[cookiemonster]

I've never heard of a "general term" before, but it seems that they want you to find an expression for the nth term of the series, i.e.

\sin{x} = \sum_{n=1}^\infty T_n

It's not difficult for the sin and cosine series. For instance, for sin(x)

T_1 = x
T_2 = -\frac{x^3}{6}

After that, they want you to use term-by-term differentiation and term-by-term integration of T_n in order to find the derivative and antiderivative of sin and cos, respectively.

cookiemonster

 

[matt grime]

General term usually just means give a formula that describes the n'th power. For sin it is, (-1)^(n)(x^2n+1)/(2n+1)! as n is 0,1,2,3...

I don't see how you could do the first part of the question without knowing this though.

 

[maccaman]

i did the first part from a formula i found on a website

the first part's formula was something like this...

cos x = 1 - (x^2 / 2!) + (x^4 / 4!) - (x^6 / 6!) - (x^8 / 8!) + ... (x^n / n!)

Can someone tell me if this is right?

 

[cookiemonster]

Would a website ever lie to you (and if so, would we ever be truthful to you)?

Not quite. You're not considering that the function skips odd powers and you're not considering the alternating sign.

cookiemonster

 

[maccaman]

yeh it had that point on the site aswell, skips odd numbers, but about the alternating sign, please do tell

 

[matt grime]

(-1)^n(x^2n)/2n!

 

[cookiemonster]

Since we all like LaTeX so much:

\cos{x} = \sum_{n=0}^\infty (-1)^n\frac{x^{(2n)}}{(2n)!}

and

\sin{x} = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}

Talk about service!

cookiemonster

 

[maccaman]

Then using the specific and General Terms, determine expressions for the derivative of sin x and the integral of cos x.

Any idea on how to start this part.

P.S. Thanks guys for helping me out

 

[cookiemonster]

Just differentiate the general term with respect to x and add all the differentiated terms together. The same for the antiderivative, except this time integrate.

cookiemonster

 

[matt grime]

There's a sign wrong in your sin term, cookiemonster. It gives -x+x^3/6...

 

[cookiemonster]

Oops.

cookiemonster

 

[maccaman]

Thanks heaps guys for all your help

 

[NSX]

Originally posted by cookiemonster
Since we all like LaTeX so much:

\cos{x} = (-1)^n\frac{x^{(2n)}}{(2n)!}

and

\sin{x} = (-1)^n\frac{x^{2n+1}}{(2n+1)!}

Talk about service!

cookiemonster

Summation signs...

:p

 

[cookiemonster]

Grr! That was a bad day.

cookiemonster

 

[jackson_sun]

This question is similar to a question i am currently undertaking in my course study...could anyone help me with the assumptions that have been assumed
Thankyou
P.S. if anyone has any ideas about the strengths and limitations of this model that would also be of great assistance

 

[jackson_sun]

sin x = x – (x3 / 3!) + (x5 / 5!) – (x7 / 7!) + (x9 / 9!) - …
‘sin x = 1 – (3x2 / 3!) + (5x4 / 5!) - (7x6 / 7!) + (9x8 / 9!) - …

cos x = 1 – (x2 / 2!) + (x4 / 4!) – (x6 / 6!) + (x8 / 8!) - …
COS x = x – (x3 / 3 / 2!) + (x5 / 5 / 4!) – (x7 / 7 / 6!) + (x9 / 9 / 8!) - …

are these the correct derivatives and integrals?

 

[Gib Z]

Do you know that the derivative of sin x is cos x? Note on your 'sin x, you have 3x^2/3!, and 5x^4/5!.

Do you know what factorial actually means? 5! = 5 times 4 times 3 times 2 times 1. Every number before it multiplied. The First term in the facotial cancels out, it give you the correct series for sin.

 

[sim17]

help!

ive found the maclaurin series for sin x and cos x , and I've tried to validate them, i even went to like the 41st term and the answer is totally wrong. I don't know wat I am doing wrong, as I am just subbing a number into x so it should be easy?

 

[Gib Z]

Are your series the correct ones? If so, check if your not putting brackets on your factions or something. Easiest one to check in sin 1, x is always just 1.

 

[sim17]

thanks for the help i worked it out, i didnt realize the greater the value of x the more terms u needed, so i was doing like sin x , x=200. haha so it was really incorrect.

hey but does anyone understand how to do this.

(for the maclaurin series obviously) find the general term (tn, where n = 1,2,3 ...) for each expression and show that each correctly generates the terms of the series.

is that just getting the specific series of sin x and cos x from the main general term??

 

[Gib Z]

General Polynomial Maclaurin series for any f(x) is

P(x)=f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3...=\sum_{n=0}^{\infty} \frac{f^n(0)}{n!}x^n

 

[sim17]

Thanks you for helping me again, ill try and finnish it from here. although stupid me, is still stuck on what to do to that equation to make it show the sin x series, in teh form

sin x = x - 1/6 x^3 + 1/120 x^5 etc...
and the cos series, cos x = 1 - 1/2 x^2 etc...