Recent questions tagged isi2020-pcb-mathematics

Deduce that if $N, H, K$ are normal subgroups of a group $G$ such that $$ N \bigcap H=N \bigcap K=H \bigcap K=\left\{e_{G}\right\} $$ and $G=H K$, then $N$ is an Abelian ...

Let $\epsilon>0$. Prove that there exists $n_{0} \in \mathbb{N}$ such that $$ n \geq n_{0} \Rightarrow 2-\epsilon<\frac{2 n+1}{n+2}<2+\epsilon. $$

Show that the sequence $\left\{x_{n}\right\}, n>0$, defined by $$ x_{n}=\int_{1}^{n} \frac{\cos (t)}{t^{2}} d t $$ is convergent.

Prove that if $N$ and $K$ are normal subgroups of a group $G$ such that $N \cap K=\left\{e_{G}\right\}$, then $x y=y x, \forall x \in N, \forall y \in K$.

Prove that the function defined by $$ f(x)=\sum_{n=0}^{\infty}\left(\frac{x^{n}}{n !}\right)^{2} $$ is continuous on $\mathbb{R}$, for any real number $x$.

Let $X_{i} \sim\left(i . i . d\right.$.) Bernoulli $\left(\frac{\lambda}{n}\right), n \geq \lambda \geq 0$.$Y_{i} \sim\left(\right. i. i. d.)$ Poisson $\left(\frac{\lambd...

Prove that if $T_{1}, T_{2}, \ldots, T_{k}$ are pairwise-intersecting subtrees of a tree $T$, then $T$ has a vertex that belongs to all of $T_{1}, T_{2}, \ldots, T_{k}$.