24.1 Let fn(x) = nx 1+2 cos² nr. Prove carefully that (fn) converges √√n uniformly to 0 on R.
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- 1. Find the F.S. expansion of f(x) = x n = x < XApply integration to the expansion 00 E(-1"x" = 1 – x +x² – x³ + ... 1+x n=0 to prove that for -1 < x < 1, (-1)"-!x" x2 x3 In(1 +x) = X -•S²³² 1 59. Show that 1 - sin x x² dx converges.Write the first five terms of p_j= 1/j and then prove p_j converges to 0.explicitly write the formula for N depending on epsilon in the first line.x2 Consider the function f(x) = + 6. 2 Calculate + 6) dæ by using the Riemann sum Rn, where the sample points are chosen to be the right endpoints of each sub-interval. Below are a couple of useful summation formulas п(п + 1) k = 2 k=1 п(п + 1) (2n + 1) 6 k=1 Rn = x2 + 6) dx = 2 NOTE: You should NOT use fundamental theorem of calculus for this question.3п 4 6п О 3п 8 O П 4. Find Dx in the Riemann Sum k=1 3п [*] 2 f(*)Ax on the interval [0, using 4 intervals.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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