Function represented by power series

Question

To what function does the function with power series , $ |x|<1$

$$F(x)=\frac{x^2}{2}-\frac{x^4}{4}+\frac{x^6}{6}-\frac{x^8}{8}+\cdots$$ converge?

Do you have some sort of a table of "famous" power series at hand (exp, log, sin, etc.)? I think you could find this series in there... — yellon, Apr 20 '15 at 16:08

score 4 · Answer 1 · answered Apr 20 '15 at 16:09

4

Differentiating, $F'(x) = x - x^3 + x^5 - x^7 + \cdots$. This is a geometric series, hence for $|x| < 1$

$$F'(x) = \frac{x}{1+x^2}$$

Now integrate to find $F$.

answered Apr 20 '15 at 16:09

Simon S

score 0 · Accepted Answer · answered Apr 20 '15 at 16:28

0

$$-F(x)=\sum_{r=1}^\infty\dfrac{(ix)^{2r}}{2r}$$

Now $\ln(1+y)+\ln(1-y)=-2\sum_{r=1}^\infty\dfrac{y^{2r}}{2r}$

answered Apr 20 '15 at 16:28

lab bhattacharjee

2 Answers2