Given that \( \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{\log \left(n^{2}+r^{2}\right)-2 ...
Channel:
Subscribers:
456,000
Published on ● Video Link: https://www.youtube.com/watch?v=Kz9E1Iz_oB8
0Given that
\( \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{\log \left(n^{2}+r^{2}\right)-2 \log n}{n}=\log 2+\frac{\pi}{2}-2 \), then \( \lim _{n \rightarrow \infty} \frac{1}{n^{2 m}}\left[\left(n^{2}+1^{2}\right)^{m}\left(n^{2}+2^{2}\right)^{m} \ldots \cdot\left(2 n^{2}\right)^{m}\right]^{1 / n} \) is equal to
(A) \( 2^{m} e^{m}\left(\frac{\pi}{2}-2\right) \)
(B) \( 2^{m} e^{m}\left(2-\frac{\pi}{2}\right) \)
(C) \( e^{m}\left(\frac{\pi}{2}-2\right) \)
(D) \( e^{2 m\left(\frac{\pi}{2}-2\right)} \)
ЁЯУ▓PW App Link - https://bit.ly/YTAI_PWAP
ЁЯМРPW Website - https://www.pw.live