Simplify expressions of real numbers and fractions



In the mathematical calculation of fractions and complicated mathematics, formulas are often considered a difficult task for students. One of the best exercises for middle and high school students is to simplify expressions of real numbers. This requires the knowledge of numbers categories and their properties. Here in this post introduce an overview of rules and properties of numbers and then apply them to compute some expressions and complicated numbers.

Show that for any $n\in\mathbb{N}$ \begin{align*}\tag{$P_n$} (n+1)!\ge \sum_{k=1}^n k!. \end{align*}

we recall that for any natural $p,$ we have $p!=1\times 2\times\cdots\times p$ and $0!=1$

Solution: For $n=1,$ we have $(1+1)!=2!\ge 1!,$ hence the property $(P_1)$ is vérified. Assume now, by reccurence, that $(P_n)$ holds. As $n+2>2,$ then \begin{align*}\tag{1} (n+2)!=(n+2)(n+1)!\ge 2(n+1)!.\end{align*}
On the other hand, by adding $(n+1)!$ to the both sides of the inequality $(P_n),$ we obatin \begin{align*}\tag{2}2(n+1)!\ge \sum_{k=1}^nk!+(n+1)!=\sum_{k=1}^{n+1}k!\end{align*}
By combining (1) et (2), we obtain \begin{align*} (n+2)!\ge \sum_{k=1}^{n+1}k!.\end{align*}
Thus $(P_{n+1})$ holds.

Let $x\in \mathbb{R}$ and $n\in\mathbb{N}^\ast$. Calculate the sum \begin{align*} S_n(x)= \sum_{k=0}^n x^k. \end{align*} Deduce the value of \begin{align*} R_n(x)=\sum_{k=0}^n k x^k. \end{align*}

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