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For over 80 years, wise investments by the US government have built up the nation's research enterprise, making it the envy of the world. Astoundingly, the Trump administration is destabilizing this enterprise by gutting funding for research, firing thousands of scientists, removing public access to scientific data, and pressuring researchers to alter or abandon their work on ideological grounds.
The undersigned are elected members of the National Academies of Sciences, Engineering, and Medicine, representing some of the nation's top scientists, engineers, and medical researchers. We are speaking out as individuals. We see real danger in this moment. We hold diverse political beliefs, but we are united as researchers in wanting to protect independent scientific inquiry. We are sending this SOS to sound a clear warning: the nation's scientific enterprise is being decimated.
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We call on the administration to cease its wholesale assault on U.S. science, and we urge the public to join this call. Share this statement with others, contact your representatives in Congress, and help your community understand what is at risk. The voice of science must not be silenced. We all benefit from science, and we all stand to lose if the nation's research enterprise is destroyed.
We demonstrated this using the series expansion for \(e^x\), and composition with the imaginary terms \(i\theta\). From that you can arrive at the most beautiful equation in all of mathematics: \[ e^{i\pi} + 1 = 0 \] which links together five of the most important constants in mathematics (\(e, i, \pi, 0, 1\)), with no extraneous fluff!
\[ |R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1} \] where \(M\) is a bound on the derivative \(f^{(n+1)}(x)\) on the interval between the center \(a\) and \(x\).
The integer case is simple: the answers are in Pascal's triangle! They're called combinations, and are defined simply in terms of factorials.
Example 6.17: