[Speedy232]

Ok so here it is. Why do the teachers give us random questions. Anyway. So Ive found the Sin x( in attachments) and Cos x(In attachments) of his formula are. How would I validate them. Like would I use a table of values... sub in degrees, pi maybe. But after I had found it the teachers asks us to use the General Term( where Tn where n = 1,2,3... ) for each expression while showing that each of them generate the series. But once you had done that ( This teacher gave us 2 weeks for this) we have to find the derivative of sin x and the integral of cos x using the specific and general terms.

I know it's a lot. But ANY help would be greatly appreciated. (Ive already used a week :( )

Sincere Speedy232

[ebaines]

Speedy - did you mean to include an attachment with your question? Without it we can't tell what your specific question is. However, if you're asking about series such as:

sin(x) = x - x^3/3! + x^5/5! - x^7/7! +...

then finding its derivative or integral isn't too difficult. The derivation of this formula comes from the Taylor series:

f(x) = f(a) + (x-a)f'(a) + (x-a)^2 f''(a)/2! +...

Set a = 0, and you get the above series.

[Speedy232]

That's odd, I swear the attachment was included, never mind that. Yes that is the formal of the sin x( that was meant to be in the attachment). So I see, you use the Taylor series to find the derivative of the sin x curves. Then what would be the derivative of the cos x series?

and to validate the sin x and cos x series. Do I just sub in any number, or what am I exactly aiming for. That's the tricky part... and then what's the general term!! Thank you for all your help.

[Speedy232]

Ok so I subbed in random numbers (pi) and I found out how many terms are needed to approx the sin... whatever


The next part is the general term. I was looking it up and it was about GP's and AP's. So I couldn't find the formula, and if I did I wouldn't be able to make it work. So if you could help me at all please I would bow down to you my master :D

[ebaines]

Sorry, I don't know what a "GP" or an "AP" is. I guess I don't qualify as your master! Can you give a bit more detail?

[Speedy232]

So it's the general formula of an a Arithmetic progressions. (AP) A-level Mathematics/AQA/MPC2 - Wikibooks, collection of open-content textbooks
Search down to AP's and you will see this

That's the general formula. Now the problem is I don't know how to use it with maclaurin series.

So you have a sin pi/2 . It works out to use 3 terms of the sin series to provide a reasonable accuracy. I said when I posted "But after i had found it the teachers asks us to use the General Term( where Tn where n = 1,2,3....) for each expression while showing that each of them generate the series" How would I use the formula for that. Any ideas?

The sine series of pi/2 comes from http://mathinsite.bmth.ac.uk/pdf/macseries_theory.pdf
Scroll down till you find the sin pi/2 example

Hope that's enough info

[galactus]

A MacLaurin series is a Taylor series centered at 0.

The series for sin(x) is:



If you differentiate this you should get the same thing as if you differentiated sin(x) because you should get the series for cos(x), which is the derivative of sin(x). See?

[ebaines]

I think by "general term" you are being asked to use the Maclauren series in its most general form:

f(x) = a0 + a1*x + a2x^2 +..

where a_n = f^n(0)/n!

So you show that if f(x) = sin(x) then you get:
a0 = 0
a1 = 1
a2 = 0
a3 = -1/3!
a4 = 0
a5 = 1/5!
etc.

Although you mention the algebraic progression (AP) series, that has nothing to do with Maclauren series. An algebraic progression is a series where each term has the same difference from the term before it, whereas a MacLauren series has smaller and smaller differences as the terms progress - that's what causes the Maclauren series to converge.

[Speedy232]

For the general term what are we using, Are we using each expression of the 5 values? Its so confusing at 2am

[ebaines]

Seems to me we've just about covered eveything already, so I don't understand what has you still so confused (even if it 2 am!). Maybe if you posted the question precisely as your teacher asked it we would better understand your confusion.

[Speedy232]

Ok here's the question: The mathematician maclaurin investigated series and found that sin x and cos x can each be written as series consisting of sum of powers and factorials. By your own research find the 2 maclaurin series validate each of them usng 5 values for x and detrermine how many tersms are needed to provide reasoable accuracy. Find the general term(tn where n=1,2,3.. Etc_ for each expression and show that each correctly generates the tersm of the series.
Then using a specific and general terms , determine expressions for the derivative of sin x and the integral of cox x. ( sorry for bad spelling)

[ebaines]

I think all this has been covered already. We've been talking about the sin(x) seris - I assume you also know what the cos(x) series is, correct? Have you tried 5 values of x to see how quickly the series converge? As a suggestion - you might try a couple with relatively small values of x (like say x = 0, or +/- pi/6, or pi/3). And a couple with much larger values - like 10*pi or 100*pi - you'll see that themagnitude of x has an affect on how quickly the series converges. As for the general term: Galactus alrady gave you a general term for the sin(x) series - you can easily write out the terms it generates for values of k = 0, 1, 2, etc. You should be able to figure out a similar general expression for the cosine series. Finally, write out the sine series to something like 6 or 7 terms (like I did for you earlier) and take the derivative of it, and you'll see that you get the cosine series. Then try integrating the terms of the cosine series and you get the sine series again. Voilà!