Uniform convergence NBHM(2019)

Question

Check the uniform convergence of following :

1) $f_{n}(x)=n\log(1+ \frac{x^{2}}{n})$ on $\mathbb{R}$

2) The series $\sum_{1}^{\infty}2^{n}$$\sin(\frac{1}{3^{n}x})$ on $[1,\infty)$

For the first question, I applied the $M_n$ test for uniform convergence but not getting the ideal value of $x$ .I am getting $x=0$,but it does't work here.

In second question, I don't have any idea which test should I use

Please help and Thank you!

Answer 1

Hints:

• $n \log\left(1+\frac{x^2}{n}\right)=\log\left(1+\frac{x^2}{n}\right)^n \to \log e^{x^2}=x^2$
• $2^{n}\sin\left(\frac{1}{3^{n}x}\right)= \frac{1}{x}\left(\frac{2}{3}\right)^n \left[\frac{\sin \left(\frac{1}{3^nx}\right)}{\frac{1}{3^nx}}\right]$ and note that $\frac{2}{3}<1$ and $\frac{1}{x}<1$