Taylor series the taylor series generated by f(x) at x=a is .. f(a) F(a) f(a)+ (x - a) + (x-( a)²+...... (x-a)n 1! 2! fn(a) n! Example/find the taylor series for f(x)=cos x at x=2π. Soltion// f(x)cos x. ƒ(2π) = 1 f(x) = -sinx. ƒ(2π) = 0 F(x) = - COS X. F(2π) = =-1 The taylor series is 1 1+0- (x- -2)²+..... 2! Example/find the taylor series for f(x)= ½- at x=2. Soltion// 1 f(x) == ½ ½ ƒ(2) = 1/1 f(x) = 11. F(2) 2 1 7(x) = 23. 7(2) = 1/13 - 11 = 8 4 = 44 The taylor series is 1 1 (x-2)+ 2 4 * 1! 1 4 * 2! (x − 2)²+..... -

e(x-3)* - 1 (x-3)² 1) The continuous function fis defined by f(x) = of all orders at x = 3. = 3. (a) Write the first four nonzero terms and the general term of the Taylor series for g(x) = e(x-3)* about x = (Hint: Use the Maclaurin series for e.) for x3 and f(3) = 0. The function fhas derivatives (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for fabout x = 3. (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for fabout x = 3 to determine whether the graph of fhas any points of inflection.

Q5/ Find Fourier series of the Function: f (x) = x if (–n < x < n)