Approximate the series of e with the finite sum M (−1)n SM(x) = Σ -(x-15)²n. 2nn! n=0 Use a graphing calculator to compare the graphs of e-(-1² with S5(x), S6(x) and S7(x) for 13 < x < (x-15)2

Using basic real analysis, solve Find the limit superior and limit inferior of each of the following sequences:

Problem 18. Find the Fourier approximation to f(x) = x² over the interval [-T, T] using the orthogonal set of vectors U₁ = 1, You may find the following integrals useful: π S u₂ = sinx, and U3 = cos x. 1 dx 2π, = π [2² Answer: f3(x) = x² dx TT [sin² zde - ² sin x dz - 0. x dx = π [² dx = 0, 7 2 = 1/37³3, [²2dx-code--4 cos² x dx = π COS x² cos x dx = -4π πT = + sin x + COST

Solve for the sum for the following:  ∞ Σ      [(-1^n)*π^(2n+1)]/[(4^(2n+1))*(2n+1)] n=0