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Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that :

$$c_0+\frac{c_1}{2}+\ldots+\frac{c_{n-1}}{n}+\frac{c_n}{n+1}=0$$

I have to prove that the equation:

$$c_0+c_1x+\ldots+c_nx^n=0$$

- Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$
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- Solving a challenging differential equation
- What is the domain of $x^x$ when $ x<0$
- Difference between Heine-Borel Theorem and Bolzano-Weierstrass Theorem
- Motivation for triangle inequality

Has a real solution between 0 and 1.

Didn’t know how to start…I thought that maybe I could use something about derivative…But I’m lost…

Any help,much appreciated!!!

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- Continuous functions with values in separable Banach space dense in $L^{2}$?
- Show uncountable set of real numbers has a point of accumulation
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- Fundamental group of torus by van Kampens theorem

**HINT 1**: $$c_0 + \dfrac{c_1}{2} + \cdots + \dfrac{c_n}{n+1} = \int_0^1 \left( c_0 + c_1x + \cdots + c_nx^n\right) dx$$

**HINT 2**:

If $f(x)$ is continuous and $\displaystyle \int_a^b f(t) dt = 0$, then can you conclude that there is a root of $f(x)$ between $a$ and $b$?

This is an exercise from the differentiation chapter in Baby Rudin, and knowledge of integration is not expected. You can solve it without integrals using the following function:

$$

f(x) = c_0x+\frac{c_1x^2}{2}+\ldots+\frac{c_{n-1}x^n}{n}+\frac{c_nx^{n+1}}{n+1}

$$

What value does it take at $0$ and $1$? What value must the derivative take in $(0, 1)$ then?

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