In this project. we use the Macburin polynomials for e x to prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s ≠ 0. 4. Write down the formula for the nth Maclaurin polynomial p n ( x ) for e x and the corresponding remainder R n ( x). Show that sn!R n (1) is an integer.
In this project. we use the Macburin polynomials for e x to prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s ≠ 0. 4. Write down the formula for the nth Maclaurin polynomial p n ( x ) for e x and the corresponding remainder R n ( x). Show that sn!R n (1) is an integer.
In this project. we use the Macburin polynomials for exto prove that e is irrational. The proof relies on supposing that e is rational and arriving a a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s ≠ 0.
4. Write down the formula for the nth Maclaurin polynomial pn(x) for exand the corresponding remainder Rn(x). Show that sn!Rn(1) is an integer.
In this project, we use the Maclaurin polynomials for f(x) = e x to proved that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose e = r/s for some integers r and s where s 6= 0. The work needs to be typed up neatly and submitted via Canvas as either a Word document or as a PDF. If you want to include pictures of your work, make sure it is neat and legible. I would prefer you use the equation editor to try to make the work as neat as possible
(1) Write the Maclaurin polynomials p0(x), p1(x), p2(x), p3(x), and p4(x) for e x . Evaluate p0(1), p1(1), p2(1), p3(1), and p4(1) to estimate e.
(2) Let Rn(x) denote the remainder when using pn(x) to estimate e x . Therefore, Rn(x) = e x − pn(x) and, in particular, Rn(1) = e − pn(1). Assuming that e = r s for integers r and s 6= 0, evaluate R0(1), R1(1), R2(1), R3(1), and R4(1).
(3) Using the results from (2), show that for each remainder R0(1), R1(1),…
In a ring, the characteristic is the smallest integer n such that nx=0 for all x in the ring. Is it acceptable to take "f" of both sides to get:
f(nx)=f(0) in the corresponding polynomial ring?
If so, is f(0) 0 in the polynomial ring?
And can we write f(nx) as nf(x)?
find the taylor polynomial of the nth degree( n=3) and x=0. then find the error estimate for the remainder. x=3
f(x) = (x+1)1/2
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
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