Question

Find Tn centered at x = 0 for all n for the function f(x) = 1 1+x (Use symbolic notation and fractions where needed.) Tn(x) = įr Consider the Taylor polynomial Tn (2) centered at x = 24 for all n for the function f(x) = 1

Answer 1

To find the Taylor polynomial Tn(x) for f(x) = 1/(1+x) centered at x = 0, calculate the n-th derivative at x = 0 and use it in the Taylor series formula, resulting in the polynomial ∑ from k=0 to n of (-x)^k.

Step-by-step explanation:

The question is asking to find the Taylor polynomial Tn(x) for the function f(x) = 1/(1+x) centered at x = 0. To find Tn(x), we need to determine the derivatives of f(x) at x = 0 and use them to construct the polynomial.

The n-th derivative of f(x) at x = 0 is given by f(n)(0) = (-1)nn!. Plugging into the Taylor series formula, we get Tn(x) = ∑k=0n (-1)kk! xk/k!, which simplifies to Tn(x) = ∑k=0n (-x)k. This is the sum of the first n+1 terms of the geometric series for 1/(1+x) when |x| < 1.