Questions from Calculus

Q: Find an infinite series that converges to the value of the given

Find an infinite series that converges to the value of the given definite integral. ∫0 1 e-x2 dx

Q: Find an infinite series that converges to the value of the given

Find an infinite series that converges to the value of the given definite integral. ∫0 1 x ex3 dx

Q: Determine all Taylor polynomials of f (x) = x4 +

Determine all Taylor polynomials of f (x) = x4 + x + 1 at x = 2.

Q: (a) Use the Taylor series for ex at x =

(a) Use the Taylor series for ex at x = 0 to show that ex > x2/2 for x > 0. (b) Deduce that e-x < 2/x2 for x > 0. (c) Show that xe-x approaches 0 as x → ∞.

Q: Let k be a positive constant. (a) Show

Let k be a positive constant. (a) Show that ekx > k2x2/2, for x > 0. (b) Deduce that e-kx 0. (c) Show that x e-kx approaches 0 as x→∞.

Q: Show that ex > x3/6 for x > 0,

Show that ex > x3/6 for x > 0, and from this, deduce that x2 e-x approaches 0 as x→∞.

Q: If k is a positive constant, show that x2e-kx

If k is a positive constant, show that x2e-kx approaches 0 as x→∞.

Q: Let Rn(x) be the nth remainder of f (

Let Rn(x) be the nth remainder of f (x) = cos x at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ |x|n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This shows...

Q: Let Rn(x) be the nth remainder of f (

Let Rn(x) be the nth remainder of f (x) = ex at x = 0. (See Section 11.1.) Show that, for any fixed value of x, |Rn(x)| ≤ e|x| * |x |n+1/(n + 1) |, and hence, conclude that |Rn(x)| → 0 as n → ∞. This...

Q: Find the Taylor series at x = 0 of the given function

Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. 1/(2x + 3)