A mathematician requires the calculation of the integral y[n] = \int_0^1 x^n exp(-x) dx where \int_0^1...

A mathematician requires the calculation of the integral
y[n] = \int_0^1 x^n exp(-x) dx

where \int_0^1 means the integral with limits 0 and 1.

A) Using integration by parts, show that y[n] satisfies the recursion y[n+1] = -1/e + (n+1) y[n]

B) Write a function that uses this recursion to calculate y[1:n] using R

@param n A postive integer
@return A vector of values y = c(y[1], ... , y[n])

Use this function to produce a vector y = c(y[1], ... , y[20]) in the object YInt

C) Recompute Yint using the pgamma and gamma functions in R.
Note that gamma(n+1)=factorial(n).

Expert Solution

A)

B)

C)

Interestingly for n>16, the recursive approach is giving the wrong results.

orchestra answered 1 year ago

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