If $\lambda\_n=\int\_{0}^{1}\frac{dt}{(1+t)^n}$ for $n\in\mathbb{N}$ , then

1.$\lambda\_n $ does not exist for some n

2.$\lambda\_n $ exists for every n and the sequence is unbounded

3.$\lambda\_n $ exists for every n and the sequence is bounded.

4.$\lim\_{x \to 0} (\lambda\_n)^{1/n}=1$

Put $t=\tan^2\theta$, therefore$\lambda\_n=\int\_{0}^{1}\frac{dt}{(1+t)^n}= 2\int\_{0}^{\pi/4} \sin\theta \cos^{2n-3}\theta d\theta$ 

How to integrate further?