If I have a denominator that goes 2 4 6 8...2k ( all multiplied) what is that pattern and how do I find it?
Similarly if I have a denominator for the odds 3 5 7...2k+1 ( all multiplied ) what is the pattern for this and how do I get it?
Thanks.
If I have a denominator that goes 2 4 6 8...2k ( all multiplied) what is that pattern and how do I find it?
Similarly if I have a denominator for the odds 3 5 7...2k+1 ( all multiplied ) what is the pattern for this and how do I get it?
Thanks.
This was the diffeq. y'' +xy' + y = 0. I have to find a power series solution. After getting a general solution, differentiating, plugging in, I get a formula. Then I have to start plugging in different values of n to see a pattern so that I can get a power series. When I start plugging in n values for the formula my denominator looks like this for the even values of a: 2*4*6*8*10
Similarly for the odd values they are 3*5*7*9.
The numerator is simple so I need not mention it.
I need a formula for these two denominators. After some research I know that for 2k the formula is 2^k(k!) and for 2k+1 it is (2k+1)! however I do no know how yet to reach that conclusion except for sheer memorization which doesn't help much.
Thanks and I hope this is clear. If not please continue to ask if you don't mind. Thank you.
I did the problem and think I see what you mean! You can write the coefficients using a number of different notational methods:
$\displaystyle a_{2n+1} = \frac{(-1)^n}{\prod_{k=1}^{k=n}(2k +1) } \, a_1
= \frac{(-1)^n \, n! \, 2^n}{(2n +1)! } \, a_1
= \frac{(-1)^n }{(2n +1)!! } \, a_1 $
and similarly
$\displaystyle a_{2n} = \frac{(-1)^n}{\prod_{k=1}^{k=n}(2k) } \, a_0
= \frac{(-1)^n }{n! \, 2^n } \, a_0
= \frac{(-1)^n }{(2n)!! } \, a_0 $ .
So the three notations used were the product notation (capital Pi), the factorial and the double factorial respectively. Hope this is of some help to you!