Study the convergence of this sequence of functions

Question

I have the following sequence of function: $$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$ and I have to study its convergence in $(0,+\infty)$. For pointwise convergence I see that for fixed $\lambda>0$ $$\lim_n f_n(\lambda)=0$$. What can a say about uniform convergence?

Answer 1

Suppose $\alpha\neq 0$ is real. Note that \begin{eqnarray*} |f_n(\lambda)|&=&\left|\frac{i}{n}\frac{1}{(\alpha-i\lambda)(\alpha-i(\lambda+\frac{1}{n})}\right|\\ &=&\frac{1}{n}\frac{1}{\sqrt{(\alpha^2+\lambda^2)(\alpha^2+(\lambda+\frac{1}{n})^2)}}\\ &=&\le \frac{1}{\alpha^2n} \end{eqnarray*} and hence $f_n(\lambda)\to 0$ uniformly as $n\to\infty$. For complex $\alpha$, you can discuss similarly.