Consider the famous Gamma function, defined by Γ(x) = R [infinity] 0 t x−1 e −t dt. Using mathematical induction, prove that Γ(n + 1) = n! for any n ∈ N ∪ {0}, where n! = n(n − 1)· · · 3 · 2 · 1 is the usual factorial function

See proof below

Step-by-step explanation:

The inductive proof consists on the following steps:

1) Base case: for n=0, we will prove that Γ(1)=0!=1. We have that

Hence the base case holds.

2) Inductive step: suppose that Γ(n + 1) = n! for some natural number n. We will prove that Γ((n + 1)+1) = (n+1)!

Use integration by parts, with the following parts:

u=t^{n+1}, du=(n+1)t^n

dv=e^{-t}, v=-e^{-t}

and we used the induction hypotheses on this last line. Also, -t^n e^-t tends to zero as n tends to infiity (the exponential decays faster than any polynomial).

We have proved the statement for n+1, and by mathematical induction, the statement holds for all n.

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have

the answer is d on edge

step-by-step explanation: