If
$\displaystyle u_n= \int_1^ne^{-t^2}dt$ where $n=1,2,3…$
Then Which of the following is true
$1)$ both the sequence $u_n$ and series $\sum u_n$ is convergent
$2)$ both the sequence $u_n$ and series $\sum u_n$ is divergent
$3)$ The sequence is convergent and $\sum u_n$ is divergent
$4)$ $\displaystyle \lim_{n \to \infty}u_n=\frac{2}{e}$
The solution I tried – I know that $\displaystyle\int_0^{\infty} e^{-t^2}=\sqrt \pi$ ,but how to calculate the $u_2,u_3…$ I have no idea please help
Best Answer
The sequence $u_n$ is convergent to $C=\int_{1}^{+\infty}e^{-t^2}\,dt$. Because of this, the series $\sum u_n$ is divergent.
No quantitative estimation is really needed, but if you want one
$$ u_n = C-\int_{n}^{+\infty}e^{-t^2}\,dt = C-\frac{1}{2}\int_{n^2}^{+\infty}\frac{dt}{e^t\sqrt{t}}\geq C-\frac{1}{2n}\int_{n^2}^{+\infty}\frac{dt}{e^t}=C-\frac{1}{2ne^{n^2}}. $$