answer:Yes, the inequality holds true for all positive integers k. We can prove this by induction. Base case (k=1): sum_{n=0}^{2} frac{sin n}{n!}>sum_{n=0}^{2} frac{cos n} {n!} which is true as shown in the original answer. Inductive step: Assume the inequality holds for some k=m: sum_{n=0}^{2m} frac{sin n}{n!}>sum_{n=0}^{2m} frac{cos n} {n!} We need to show that it also holds for k=m+1: sum_{n=0}^{2(m+1)} frac{sin n}{n!}>sum_{n=0}^{2(m+1)} frac{cos n} {n!} Expanding the sums and using the inductive hypothesis, we get: begin{align*} sum_{n=0}^{2(m+1)} frac{sin n}{n!} &>sum_{n=0}^{2(m+1)} frac{cos n} {n!} Rightarrowqquad sum_{n=0}^{2m} frac{sin n}{n!}+frac{sin (2m+1)}{(2m+1)!}+frac{sin (2m+2)}{(2m+2)!}&>sum_{n=0}^{2m} frac{cos n} {n!}+frac{cos (2m+1)}{(2m+1)!}+frac{cos (2m+2)}{(2m+2)!} Rightarrowqquad frac{sin (2m+1)}{(2m+1)!}+frac{sin (2m+2)}{(2m+2)!}&>frac{cos (2m+1)}{(2m+1)!}+frac{cos (2m+2)}{(2m+2)!} Rightarrowqquad frac{sin (2m+1)-cos (2m+1)}{(2m+1)!}+frac{sin (2m+2)-cos (2m+2)}{(2m+2)!}&>0 end{align*} Since 0<2m+1<frac{pi}{2} and frac{pi}{2}<2m+2<pi, we have sin (2m+1)-cos (2m+1)>0 and sin (2m+2)-cos (2m+2)>0. Therefore, the inequality holds for k=m+1. By induction, the inequality holds for all positive integers k.

answer:To differentiate the given function, we can use the sum rule and the power rule. The sum rule states that the derivative of a sum of functions is equal to the sum of the derivatives of each function. The power rule states that the derivative of x^n is equal to nx^{n-1}. Using these rules, we can differentiate the given function as follows: frac{d}{dx}[sqrt[3]{6-2x} - tan(2-8x)] = frac{d}{dx}[sqrt[3]{6-2x}] - frac{d}{dx}[tan(2-8x)] = frac{-2}{3sqrt[3]{(6-2x)^2}} - 8sec^2(2-8x) Therefore, the derivative of the given function is frac{-2}{3sqrt[3]{(6-2x)^2}} - 8sec^2(2-8x). The answer is frac{-2}{3sqrt[3]{(6-2x)^2}} - 8sec^2(2-8x)

answer:1. Let {v_1} = {-1,0,2} 2. Normalize {v_1} to obtain {u_1} = left{-frac{1}{sqrt{5}},0,frac{2}{sqrt{5}}right} 3. Let {v_2} = {2,-1,1} - Subtract the projection of {v_2} onto {u_1} from {v_2}: {v_2} - left(frac{{v_2} cdot {u_1}}{{u_1} cdot {u_1}}right){u_1} = {2,-1,1} - left(frac{2}{5}right)left{-frac{1}{sqrt{5}},0,frac{2}{sqrt{5}}right} = left{frac{11}{5},-frac{1}{sqrt{5}},frac{3}{5}right} - Normalize the result to obtain {u_2} = left{sqrt{frac{2}{3}},-frac{1}{sqrt{6}},frac{1}{sqrt{6}}right} 4. Let {v_3} = {-1,1,2} - Subtract the projections of {v_3} onto {u_1} and {u_2} from {v_3}: {v_3} - left(frac{{v_3} cdot {u_1}}{{u_1} cdot {u_1}}right){u_1} - left(frac{{v_3} cdot {u_2}}{{u_2} cdot {u_2}}right){u_2} = {-1,1,2} - left(frac{1}{5}right)left{-frac{1}{sqrt{5}},0,frac{2}{sqrt{5}}right} - left(frac{4}{3}right)left{sqrt{frac{2}{3}},-frac{1}{sqrt{6}},frac{1}{sqrt{6}}right} = left{-frac{4}{15},frac{11}{6},frac{1}{sqrt{30}}right} - Normalize the result to obtain {u_3} = left{sqrt{frac{2}{15}},sqrt{frac{5}{6}},frac{1}{sqrt{30}}right} Therefore, the orthogonalized vectors are: {u_1} = left{-frac{1}{sqrt{5}},0,frac{2}{sqrt{5}}right} {u_2} = left{sqrt{frac{2}{3}},-frac{1}{sqrt{6}},frac{1}{sqrt{6}}right} {u_3} = left{sqrt{frac{2}{15}},sqrt{frac{5}{6}},frac{1}{sqrt{30}}right}

answer:Experiments involving special relativity on Earth often neglect general relativistic effects because the gravitational field is weak. However, these approximations do introduce errors. The equivalence principle (EP) asserts that in a small, freely falling laboratory, spacetime is locally flat, and thus special relativity's predictions hold true. This has been confirmed through experiments, like the Eötvös experiment, which tests the equality of inertial and gravitational mass but doesn't specifically test general relativity. When the experiment is conducted in a non-free-falling setup, like one anchored to the Earth's surface, the EP predicts results similar to a rocket accelerating at g. The Pound-Rebka experiment is an example of this, verifying SR combined with the EP but not specifically GR. To directly test GR near the Earth's surface, experiments must involve a spatial region large enough to have varying accelerations. Such tests include the Hafele-Keating experiment, which studied time dilation in moving clocks, and the detection of gravitational waves by LIGO and Virgo, which are sensitive to the minute distortions of spacetime caused by these waves. These experiments can detect deviations beyond what SR predicts, thus confirming aspects of general relativity.