l'hopital and taylor series questions
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Find lim┬(x→0+)〖x^x 〗. (personal note: I think the approach is to use l'Hopital's rule)
Prove that e is irrational. (hint: Suppose false, so that e = p/q where p,q ∈ N. Write e = e^1= P_n (1)+ R_n (1), multiple both sides by n! and deduce a contradiction when n ∈ N is sufficiently large).
Expand the polynomial p(x)= 〖3x〗^3+〖2x〗^2- x+1 as a polynomial in powers of (x-1): That is, show that p(x)= ∑_(k=0)^3▒c_k 〖(x-1)〗^k and find the values of the constants c_(0 ),...,c_3
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1. Find . (personal note: I think the approach is to use l'Hopital's rule)
Solution:
Let y =
Take log on each side
lny = ln( )
lny =
We can write it as
lny =
Now apply L'Hospital's rule
lny =
lny tends to 0.
Therefore y will tend to e^0 = 1.
2. Prove that e is irrational. (hint: Suppose ...
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