Find the sum of the series. 3+9/2!+27/3!+81/4!+. . .

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence.

$$f(x)=x-x^3, a = -2$$

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) to 0.] Also find the associated radius of convergence. f(x)= sin pi x

Find the Maclaurin series f(x) for using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x)→0.] Also find the associated radius of convergence. f(x) = sinh x

Evaluate the indefinite integral as an infinite series. integral cosx-1/x dx

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x)→0.] Also find the associated radius of convergence. f(x) = ln(1+x)

Find the Taylor series for f centered at 4 if f^(n)(4)=(-1)^n*n!/3^n(n+1)

Use the Maclaurin series for e^x to calculate 1/^10√e correct to five decimal places.

Find the Taylor series for f(x) centered at the given value of a. [Assume that t has a power series expansion. Do not show that Rn(x)→0.] Also find the associated radius of convergence. f(x) = sin x, a = π/2

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) to 0.] Also find the associated radius of convergence. f(x)=x^1/2, a=16

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? summation n=1 to infinity (-1)^n-1/n!

Approximate f by a Taylor polynomial with degree n at the number a.

$$f(x)=x^2/^3, a=1, n=3, 0.8≤x≤1.2$$

What is wrong with the following calculation? 0 = 0 + 0 + 0 + ... = (1-1)+(1-1)+(1-1)+... =1-1+1-1+1-1+... =1+(-1+1)+(-1+1)+(-1+1)+... =1+0+0+0+...=1 (Guido Ubaldus thought that this proved the existence of God because “something has been created out of nothing.”)

What is a conditionally convergent series?