Question

Salut, am nevoie de ajutor la ex. 1.252C

Answer 1

$S = \dfrac{1}{\sum\limits_{k=1}^n \log_{2}k}+\dfrac{1}{\sum\limits_{k=1}^n \log_{3}k}+...+\dfrac{1}{\sum\limits_{k=1}^n \log_{n}k} \\ \\ S = \dfrac{1}{\sum\limits_{k=1}^n\dfrac{\ln k}{\ln 2}}+ \dfrac{1}{\sum\limits_{k=1}^n\dfrac{\ln k}{\ln 3}}+...+ \dfrac{1}{\sum\limits_{k=1}^n\dfrac{\ln k}{\ln n}}\\ \\ S = \dfrac{1}{\dfrac{1}{\ln 2}\sum\limits_{k=1}^n\ln k}+\dfrac{1}{\dfrac{1}{\ln 3}\sum\limits_{k=1}^n\ln k}+...+\dfrac{1}{\dfrac{1}{\ln n}\sum\limits_{k=1}^n\ln k}$

$S =\dfrac{\ln 2}{\sum\limits_{k=1}^n \ln k}}+ \dfrac{\ln 3}{\sum\limits_{k=1}^n \ln k}}+...+\dfrac{\ln n}{\sum\limits_{k=1}^n \ln k}}$

$S = \dfrac{1}{\sum\limits_{k=1}^n \ln k}} \cdot\Big(\ln2+\ln 3+...+\ln n\Big) \\ \\\\S =\dfrac{1}{\sum\limits_{k=1}^n \ln k}}\cdot \sum\limits_{k=1}^n \ln k}\\ \\ \\\Rightarrow \boxed{S = 1}$