\varphi summation

Let \varphi denote the Euler’s function. Prove that

    \[\sum_{k=1}^{\infty} \varphi(k) \left \lfloor \frac{n}{k} \right \rfloor = \frac{n \left ( n+1 \right )}{2}\]

Solution

The key idea is to rewrite the floor as a sum involving divisors. Hence,

    \begin{align*} \sum_{k=1}^{\infty} \varphi(k) \left \lfloor \frac{n}{k} \right \rfloor &= \sum_{k=1}^{\infty} \varphi(k) \sum_{\substack{m \leq n \\ k \mid n}} 1 \\ &= \sum_{k=1}^{\infty} \sum_{\substack{m \leq n \\ k \mid n}} \varphi(k) \\ &= \sum_{m=1}^{n} \sum_{k \mid m} \varphi(k) \\ &= \sum_{m=1}^{n} m \\ &= \frac{n \left ( n+1 \right )}{2} \end{align*}

Read more

Equal segments

Let f(x) = \frac{1}{x} and let (\varepsilon) be a line that intersects \mathcal{C}_f at the points \mathrm{A} , \Delta and the axis at the points \mathrm{B} and \Gamma.

Rendered by QuickLaTeX.com

Prove that \mathrm{AB}= \Gamma \Delta.

Solution

Let (\varepsilon):y= \alpha x + \beta with \alpha \neq 0. Then,

    \[\alpha x + \beta = \frac{1}{x} \Leftrightarrow \alpha x^2 + \beta x - 1 = 0 \Leftrightarrow \left\{\begin{matrix} x_1 + x_2 &= & - \dfrac{\beta}{\alpha} \\\\ x_1 x_2 & = & - \dfrac{1}{\alpha} \end{matrix}\right.\]

Hence,

    \begin{align*} \overrightarrow{\mathrm{BA}} &= \left ( x_1, \frac{1}{x_1} - \beta \right ) \\ &= \left ( - \frac{\beta}{\alpha} -x_2, - \frac{1}{x_2} \right ) \\ &= \overrightarrow{\Delta \Gamma} \end{align*}

Read more

Integral inequality (I)

Let f:[0, 1] \rightarrow \mathbb{R} be a differentiable and convex function such that f(0)=0 and f'(1)= 2. Prove that \int_{0}^{1} f(x) \, \mathrm{d}x < 1.

Symmetry

Let f, g : \mathbb{R} \rightarrow \mathbb{R} be continuous functions. If \mathcal{C}_f is symmetric around the line x=\frac{\alpha + \beta}{2} then prove that:

    \[\int_{\alpha}^{\beta} x g \left ( f(x) \right )\, \mathrm{d}x = \frac{\alpha+\beta}{2} \int_{\alpha}^{\beta}g \left ( f(x) \right )\, \mathrm{d}x\]

An integral from a geometric view

Evaluate the integral

    \[\mathcal{J} = \int_0^1 \sqrt{4-x^2} \, \mathrm{d}x\]

Solution

 

Read more

Donate to Tolaso Network