Question

Salut, ma puteti ajuta la problema numarul 675, eu am ajuns aici...Nu stiu ce sa fac cu acel 1, ma puteti ajuta ?

Answer 1

Răspuns

Explicație pas cu pas:

$\displaystyle \limit\lim_{n\to\infty} \left(n-\sum_{k=1}^n e^{\frac{k}{n^2}}\right)=\displaystyle \limit\lim_{n\to\infty} \sum_{k=1}^n\left(1-e^{\frac{k}{n^2}}\right)=\\ \displaystyle \limit\lim_{n\to\infty} \sum_{k=1}^n\left(\dfrac{1-e^{\frac{k}{n^2}}}{\frac{k}{n^2}}\cdot \dfrac{k}{n^2}\right) = -\limit\lim_{n\to\infty} \sum_{k=1}^n \dfrac{k}{n^2} = -\limit\lim_{n\to\infty} \dfrac{\sum_{k=1}^n k}{n^2} = \\=-\limit\lim_{n\to\infty} \dfrac{n(n+1)}{2n^2} = \bold{-\dfrac{1}{2}}\\Raspunsul~este~E.$