21 May 04
Originally posted by geniusWell a general law for a quiz type thread is that the originator knows the answer to the question. Since you confirmed to us that you got it right. I induce that it must be proven. 😀
prove by induction that ((d^n)/(d(x^n)))/(x(e^x))=(x+n)(e^x) for all integers n>=1.
(incase you're sondering-this is the other part of my paper i couldn't get...however, i have recently discovered that i have got it right...apparently...🙂 tis a 5-marker...)
22 May 04
Originally posted by geniusI suppose that you mean the n-th derivative to x, and you typed a '/' too many?? I will write E for e^x and D for derivative or d/dx
prove by induction that ((d^n)/(d(x^n)))/(x(e^x))=(x+n)(e^x) for all integers n>=1.
(incase you're sondering-this is the other part of my paper i couldn't get...however, i have recently discovered that i have got it right...apparently...🙂 tis a 5-marker...)
D^n (xE) = (x+n)E
Case n=1: D^1 (xE) = xE + E = (x+1)E, correct
INDUCTION STEP: Suppose the statement is correct for n=k, then D^(k+1) (xE) = D^1 ( D^k (xE)) = D^1 ((x+k)E) = (x+k)E + E = (x+k+1)E, correct
Thus the given statement is correct for all integers n >= 1