Question

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(a) Estimate the Taylor series and its radius of convergence for the functions:

(i) sin (x)
(ii) cos (x)
(iii) eˣ

Answer 1

The Taylor series expansion for the functions sin(x), cos(x), and e^x, and their respective radii of convergence.

Step-by-step explanation:

The Taylor series expansion of a function is a way to approximate the function by using a polynomial.

(i) For the function sin(x), the Taylor series expansion is:
`sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + ...` which means the series converges for all values of x.

(ii) For the function cos(x), the Taylor series expansion is:
`cos(x) = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + ...` which means the series converges for all values of x.

(iii) For the function e^x, the Taylor series expansion is:
`e^x = 1 + x + (x^2/2!) + (x^3/3!) + ...`nite, which means the series converges for all values of x.