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- (b) A sequence (fn) of differentiable functions such that (fn ) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.b) Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many terms of (xn) lie outside of the interval (−e ,e).The claim is.Proof of answer:(c) A sequence (fn) of differentiable functions such that both (fn) and (fn )converge uniformly but f = lim fn is not differentiable at some point
- 1. Let {gn} be a sequence of integrable functions which converges a.e. to an integrable function g. Let {fn} be a sequence measurable functions such that fn < In and {fn} converges to f a.e. Also suppose that lim f gn = S g. Prove that f |fn-f|→ 0.1.)Determine whether the Fourier series of the following functions converge uniformly or not. a. f(x) = ex, - 1 < x < 1 b. f(x) = sinh(x), - π< x < π Answer: a)The periodic function is not continuous at ± etc., so convergence cannot be uniform. b. Like a, the periodic function has jumps at ±π etc.1. Let fgng be a sequence of integrable functions which converges a.e. to an integrable function g. Let ffng be a sequence of measurable functions such that jf j g and ff g converges
- 1.)Determine whether the Fourier series of the following functions converge uniformly or not. Sketch each functions a. f(x) = sin(x) + |sin(x)|, - π < x < π b. f(x) = x + |xl, π < x <π Answer: a. This periodic function is continuous, and its derivative is continuous except for jumps at ± π etc. Convergence is uniform. b. The periodic function has jumps, so convergence cannot be uniform.Find the series of representation for f(x)=ln(14-x)The Bessel function of order 0 is represented by the series J0(x) = sum of x=∞ and n=0. (−1)^ nx^2n/2^2n(n!)^2. Find the derivative of the Bessel function.