Does fn(x)=cosn(x)(1−cosn(x))f_n(x)=\cos^n(x)(1-\cos^n(x)) converge uniformly for xx in [π/4 , π/2][π/4 , π/2]?

Does f_n(x)=\cos^n(x)(1-\cos^n(x)) converge uniformly for x in [π/4 , π/2]? Its clear to see that the point-wise convergence is to 0. By finding the derivative I obtained that the maximum of f_n is when \cos^n(x)=1/2 and that \sup |f_n(x)-0|= 1/2\cdot(1-1/2)=1/4 which would indicate that the function does not converge uniformly, however I’m not sure the … Read more

Show $\sum_{n=1}^\infty \frac{\mathrm{ln}(n)}{n^x}$ is not uniformly convergent on $(1,\infty)$.

Show $\displaystyle\sum_{n=1}^\infty \frac{\mathrm{ln}(n)}{n^x}$ is not uniformly convergent on $(1,\infty)$. I thought I can show that $\displaystyle\sup_{x\in (1,\infty)}\left|\frac{\mathrm{ln}(n)}{n^x}\right|\not\to 0$, but it does, so this doesn’t help. Answer Use the fact that the series diverges at $x = 1$. Rewrite the series as $$f(x) := \sum_{n = 1}^{\infty} \frac{\ln n}{e^{x \ln n}}$$ So the error term can … Read more

Uniform convergence of power series ∑∞n=1xn(n+1)(n+2)\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}

Prove the uniform convergence of power series ∞∑n=1xn(n+1)(n+2) on the closed interval [−1,1]. The radius of convergence R=limn→∞|anan+1|=limn→∞|(n+2)(n+3)(n+1)(n+2)|=1 But how do you prove it is uniformly convergent on [-1,1] ? Answer Hint. From the inequality \left|\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}\right|\leq \sum_{n=1}^\infty \frac 1{(n+1)^2}<\infty one deduces the uniform convergence of the series on [-1,1]. AttributionSource : Link , … Read more

Uniform convergence on every bounded closed intervals implies uniform convergence on $\Bbb R$

When I read my lecture notes, I found that the outline of the proof for uniform convergence of cosine function$$\cos x =1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots$$on $\Bbb R$ is as the following: On $[-M,M]\ (M>0)$, the cosine function is uniformly convergent on $[-M,M]$ by M-Test. Since $M$ is arbitrary, it follows that it is uniformly convergent on $\Bbb R$. … Read more

Do these functions converge uniformly?

Let n∈N. Consider the function fn:R→R:x↦1−3n(x−1+1n)2+xn. For all x∈R we have limn→∞fn(x)=1. I need to determine whether the convergence is uniformly or not. I think it is not uniformly convergent. I tried to prove this, but I’m having trouble finding the correct x and the correct estimate. We need to prove that ∃ϵ>0,∀n0∈N,∃x∈R,∃n≥n0:|fn(x)−1|≥ϵ. I let … Read more

Example of a series of functions that converges absolutely but does not converges uniformly?

Can someone suggest an example of a series of functions that converges absolutely but does not converges uniformly? Answer Let fn(x)=xn. ∑fn(x) is absolutly convergent at [0,1) as a geometric one. it doesn’t converge uniformly at [0,1) cause sup. AttributionSource : Link , Question Author : Sørën , Answer Author : hamam_Abdallah

Pointwise Convergence. Uniform Convergence

would really appreciate help in understanding pointwise convergence and uniform convergence. For example for the questions below, how does f_n converge pointwise to f(x)=0? To see whether a function converges pointwise or not, don’t you just take the limits of the sequence of functions f_n. Because wouldn’t lim(f_n(x))=infinity rather than than 0? Could someone please … Read more

Check the uniform convergence of series of functions

Check that given series converges uniformly for x∈(−π,π) or not 1) ∑∞n=1(xn)n 2) ∑∞n=11((x+π)n)2 first series is power series so I find the radius of convergence that comes R=e which is inside the given set so I am not able to give the final conclusion. after this I were thinking to apply Mn test , … Read more

can we conclude that f2n→f2f_n^2\rightarrow f^2 uniformly on (−∞,∞)(-\infty, \infty)?

Assume (fn) is a sequence of differentiable function. If fn→f uniformly on (−∞,∞), can we conclude that f2n→f2 uniformly on (−∞,∞)? If not, is there any counterexample? I don’t think this is true but having trouble of finding a counterexample. |f2n−f2|=|(fn+f)(fn−f)|, so if it’s uniformly convergent, the distance should be converging to 0 depending on … Read more

Uniform Convergence of \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+…\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+…

Show that following series is uniformly convergent in (-1,1). \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+… I had previously solved this problem using Weierstrass M test by taking |f_n| = |\frac{2^n.x^{2^n-1}}{1+x^{2^n}}| \leq 2^n.k^{2^n-1}, k\in(x,1) Thus, \frac{M_n}{M_{n+1}}\to \infty hence the series is uniformly convergent. But now I think this is incorrect as k shouldn’t be dependent on x. How should I proceed … Read more