Which of the following functions are continuous$?$

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(i)$ f(x)=x^2+\frac {x^2}{1+x^2}+\frac {x^2} {({1+x^2})^2}+\cdots,x\in\mathbb R$

(ii)$f(x)=\displaystyle\sum_{n=1}^\infty(-1)^n\frac{\cos nx}{n^{\frac 3 2}}\,,x\in[-\pi,\pi]$

(iii)$f(x)=\displaystyle\sum_{n=1}^\infty n^2x^n\,,x\in\left[-\frac 12,\frac 1 2\right]$

MY TRY:(ii)and(iii) look like sequence of function but the question is about continuity .So I am just clueless.Thank you

Note:ans:(ii)and(iii) are continuous.

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1 Answer

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We have

$$\forall n>0 \;\;\forall x\in[0,\pi]\;\;$$

$$ |(-1)^n\frac{\cos(nx)}{n^{\frac{3}{2}}}|\leq \frac{1}{n\sqrt{n}}$$

$\frac{3}{2}>1\implies $ the series of functions converges normlly and uniformly at $I=[0,\pi].$

the functions $x\mapsto (-1)^n\frac{\cos(nx)}{n\sqrt{n}}$ are continuous at $I$ thus the function sum is continuous at $I$.

for the third, by ratio test, the radius of convergence is $1$, and the function sum of a power series is $C^{\infty}$ inside $(-1,1)$.

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