Category Archives: Algebra

Let $a_n=\sum_{d|n}\dfrac{1}{2^{d+\frac{n}{d}}}$. In other words, $a_n$ is the sum of $\dfrac{1}{2^{d+\frac{n}{d}}}$ over all divisers $d$ of $n$. Find $$\dfrac{\sum_{k=1}^{\infty}ka_k}{\sum_{k=1}^{\infty}a_k}=\dfrac{a_1+2a_2+3a_3+…}{a_1+a_2+a_3+…}$$ Click here for the solution. Solution: For the denominator, we have: $$\begin{align} \sum_{n=1}^{\infty}a_n & = \sum_{n=1}^{\infty}\sum_{d|n}\dfrac{1}{2^{d+\frac{n}{d}}} = \sum_{d=1}^{\infty}\sum_{n\ge1,d|n}\dfrac{1}{2^{d+\frac{n}{d}}} \\ & = … Continue reading →

The Telescopic Method is a technique for calculating the summary or product of a certain series in which each term can be decomposed into multiple parts, with some of them cancelling those of the next term. For example, to calculate … Continue reading →

Let $a$ and $b$ are positive integers that makes $\dfrac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ a rational number, where $a <= b$. Find all pairs of $(a, b)$.