Question

write the following functions as power series, and calculate their open intervals of convergence
c) \(\log(1+3x^{2})\)

Answer 1

If by log you mean ln, then
\[\begin{align}
\ln(1+x)=\sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} &\text{for }|x|<1 \\
\ln(1+3x^2)=\sum_{n=0}^\infty (-1)^{n+1} \frac{(3x^2)^n}{n} &\text{for }|3x^2|<1 \Rightarrow |x|<\frac{1}{\sqrt{3}}\\
\end{align}\]
Therefore, the R of the new series is \(\large \frac{1}{\sqrt3}\).

Answer 2

ok thank you very much blockcolder, its much apreciated to me, i will let you the know the point take from questions if you will be here next week , now i am goint to university to give homework, take care