Series/binomial/multinomial theorum

  • Thread starter roadrunner
  • Start date
In summary, for any positive integer n, we need to determine the sum of (-1)^i * i!(n-i)! from i=0 to n and the sum of 1/i!(n-1)! from i=0 to n. The binomial expansion of (1+x)^n can be used to solve this problem, with x=1 or -1. The solution for a) is 2^n/n!, while the solution for b) is 0.
  • #1
roadrunner
103
0

Homework Statement



for any positive interget n determine

a) SUM(from i=0 to n) of

(-1)^i
i!(n-i)!

b) SUM (from i=0 to n) of

1
i!(n-1)!

Homework Equations





The Attempt at a Solution



a) well i didnt realyl know how to start. i found that with
n=1 then it becomes 1+1
n=2 .5+1+.5
n=3 1/3! + .5 +.5 +1/3!
n=4 1/4! +1/3! +.25 +1/3! +1/4!
n=4 1/5! +1/4! +.083333 +.083333 +1/4! +1/5!

so i guess what i need is how do i get the middle term?
or am i going about it wrong?

for b) i know the sum is always 0, because ichecked it on my calculator. but i don't know hwo to prive that
 
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  • #2
Do you know what the binomial expansion of (1+x)n is?
 
  • #3
i acctulay think i got it
a) 2^n/n!
 
  • #4
Yes. Do the binomial expansion of (1+x)n, and the set x=1 or -1 to get the two versions of your problem.
 
  • #5
so is the 2^n/n! right?
i get the same answer doing that as i do doing it long form (calculator)

(mind helping on my other post ? :P)
 

1. What is the series/binomial/multinomial theorem?

The series/binomial/multinomial theorem is a mathematical formula that describes the expansion of a binomial or polynomial raised to a power. It allows us to calculate the coefficients of each term in the expanded form.

2. How is the series/binomial/multinomial theorem used in real life?

The series/binomial/multinomial theorem is used in various fields such as statistics, physics, engineering, and economics. It is used to simplify complex equations and to solve problems involving probability, combinations, and permutations.

3. What is the difference between the binomial and multinomial theorem?

The binomial theorem is used to expand a binomial expression raised to any power, while the multinomial theorem is used to expand a polynomial expression raised to any power. The binomial theorem has two terms, whereas the multinomial theorem can have multiple terms.

4. What are the applications of the series/binomial/multinomial theorem?

The series/binomial/multinomial theorem has many applications in mathematics, including solving problems involving probability, combinatorics, and calculus. It is also used in fields such as physics, engineering, and economics, to simplify complex equations and solve real-world problems.

5. How can I remember the formula for the series/binomial/multinomial theorem?

One way to remember the formula is by using the acronym FOIL, which stands for First, Outer, Inner, Last. This reminds us of the pattern in which the terms are multiplied when expanding a binomial or polynomial expression. Another way is to practice using the formula in different problems to become more familiar with it.

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