Which of the following are true
- $\sum_{n=1}^{\infty}\frac{(-1)^n + \frac{1}{2}}{n} $ is convergent.
I have tried by Lebnitz rule., but $|a_n|$ is not a decreasing sequence, So i can not use this, i have also tried to prove the series absolutely convergent, but not getting any sucees.
2.The series $$\sum_{n=1}^{\infty} \frac{x^2}{1+ n^2x^2}$$
converges uniformy on $\mathbb R$
I have tried by Weierstrass M-test, but $f_n(x) = \frac{x^2}{1 + n^2x^2}$ is unbounded on $\mathbb R$.Is i conclude that if $f_n(x)$ is unbounded , then $\sum f_n(x)$ is not uniformly convergent., but it is pointwise convergent.
Please tell me how to proceed.
- The series $$\sum_{n=1}^{\infty}\frac{\sin nx^2}{1+n^3}$$
converges uniformly on $\mathbb R$
Here $ |f_n(x)| \leq M_n = \frac{1}{1 + n^3}$ and $\sum M_n$ is convergent , then $\sum_{n=1}^{\infty} f_n(x)$ is unformly convergent by Weierstrass M-test.
Please tell me about (1) and (2) and check the (3) option.
Any help would be appreciated. Than you
$\frac{x^2}{1+n^2x^2}<\frac{1/n^2+x^2}{1+n^2x^2}=1/n^2$ $\frac{（-1)^n+\frac{1}{2}}{n}=\frac{（-1)^n}{n}+\frac{1}{2n}$
$\frac{（-1)^n}{n}$‘s series is convergent,while $\frac{1}{2n}$’s series is +∞