Leaving Cert Maths Q.

ZorbaTehZ

Paper 2, question 8 - the part b's usually include a Maclaurin Series(MlS), followed by a part where you have to write out a general term (U/subset(n)) for a sum to infinity of the MlS you just did.
The problem I've had for a while is trying to determine if its a sum to infinity for n=0 or n=1, since in some cases it's latter, and in some it's the former (when I say n=0, I mean the n=x below the sigma sign - the initial value)
Maclaurin Series (you can see some examples of this mid-way down the page)

For example(s):

1. Write out the MlS for f(x)=e^x
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

In this case the general term is:
U/subset(n) = (x^2n)/(2n!)

2. Write out the MlS for f(x)=Sin x
Sin x = x/1! - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...

In this case the general term is:
U/subset(n) = (-1)^(n-1) . [x^(2n-1)]/(2n-1)!

You can see that the first example is a sum to infinity where initial value of n is 0, and in the second case the initial value of n is 1.
I have problems trying to determine which it is in each case, since I need to know to be able to write out the general term.

Conclusion I've come to is that if the the first term of the MlS is zero then its a sum to infinity for n=1, but when the first term is non-zero (like the 1 at the start of the e^x) then its a sum to infinity where n=0.

Hoping someone could tell me if this is right, since I asked my school-teacher about it and he hadn't a clue.

Farouk.Bulsara

ZorbaTehZ wrote:

For example(s):

1. Write out the MlS for f(x)=e^x
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

In this case the general term is:
U/subset(n) = (x^2n)/(2n!)

First of all, the general term in the series for e^x is

U_n = x^n / n!

and not as you have written above. Otherwise you'd miss all the "odd" terms in the series.

Secondly, you can write the general term starting at any integer really, for example:

U_n = x^(n+1) / (n+1)! where n starts at -1
U_n = x^(n-1) / (n-1)! where n starts at +1
U_n = x^(n-42) / (n-42)! where n starts at +42

Starting a Taylor/Maclaurin series at 0 or 1 is just a matter of taste, or force of habit, or experience, or whatever. The end result is just the same.

Fred

ZorbaTehZ

Aye, my mistake.
That Un I wrote down is for (e^x + e^-x)/2 was looking at the wrong place.

About that second part you wrote, if I was going to use a ratio test where you need Un+1/Un, does it still not matter? Maybe I'm just doing the questions wrong but I keep getting wrong answers.

ZorbaTehZ

In the test they're going to ask for a specific Un - not a Un with a summation sign, where n is defined - just a Un, so that approach might be correct but it doesn't work tbh in the LC test.

Fremen

The ratio test still works for the example using 42:

U_n = x^(n - 42)/(n - 42)!
U_(n + 1) = x^(n - 41)/(n - 41)!

so (U_n) / (U_(n + 1)) = (x^(n - 42))*((n - 41)!) / ((x^(n - 41) *(n - 42)!)

= x/(n - 42) I hope

letting n -> infinity, the limit is zero, so the series converges (I hope. Haven't used the ratio test in five years).

ZorbaTehZ

Thanks Fremen.
However, it's the Un thats bothering me. I think I will repost this in the LC forum, since the students might know better, which it is.

Fremen

yeah, you might get better advice there.
U_n just means "the n-th term", as in, put any number n in there and you'll get a term in the sequence

ZorbaTehZ

Actually if you look at that link in my op, you can see what I mean if you look down at the e^x and compare it with Log(1+x)

Maybe you can explain to me why he defined one as n=0 and the other as n=1?

That Un that is written beside the sums is actually exactly the thing that I need to be able to reproduce in an exam. Thats how come I'm confused with when I need to use n=0 or n=1.

Fallen Seraph

I believe that the answer is simply choice.

One can also write the series for sin x as

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

for a sum from n=0 and it returns exactly the same infinite sum. The two statements are equivalent.

ZorbaTehZ

Well whatever method he using, it's exactly the same as all the General Terms that are asked in the LC. If I want to reproduce the series as defined on that page, the method I described above seems to work for them all so I'll take it that it's right.

Fremen

Hmm, I got your PM but I'm not following this thread too well. Is it writing out the general term that you're having difficuity with?

Normally, you would just pick your starting number so that the general term is reasonably easy to understand at a glance. That's why you start e^x at 0.

Really, to get the n-th term, it's just a matter of writing out the first few terms in the sequence, (i.e. f(0), f'(0)x, f''(0)x^2/2!) and spotting a pattern. Then generalise the pattern so it uses N's instead of numbers, and sum from whatever number corresponds to the start of the sequence

Be sure to specify what number you're beginning your summation from, though.

I'm not sure if that's the info you were looking for, or what.