I want to find the limit of $\left(1+\frac{1}{n^{2}} \right)\left(1+\frac{2}{n^{2}} \right)...\left(1+\frac{n}{n^{2}} \right)$ as $n \rightarrow \infty$, any hint?
Asked
Active
Viewed 111 times
-1
-
1See: http://math.stackexchange.com/questions/2022413/is-the-sequence-a-n-prod-limits-i-1n-left1-fracin2-right-decre ; http://math.stackexchange.com/questions/389155/how-to-calculate-lim-n-to-infty11-n212-n2-cdots1n-n2 ; http://math.stackexchange.com/questions/183061/how-to-evaluate-lim-limits-n-to-infty-prod-limits-k-1n-1k-n2 ; http://math.stackexchange.com/questions/904260/what-is-the-limit-of-the-following-sum – Winther Jan 09 '17 at 06:13
2 Answers
2
HINT: Logarithm of it is equal to $\frac{1}{n^2}(1+2+\cdots+n)+o(1/n^2)$.
Przemysław Scherwentke
- 13,668
- 5
- 35
- 56
-
@Winther Oh, I see, $o(1/n^2)$ in memory... Correcting, – Przemysław Scherwentke Jan 09 '17 at 06:35
0
Hint When $n$ gets large, each of the factors $(1+k/n^2)$ is approximately $e^{k/n^2}.$
spaceisdarkgreen
- 58,508