Calculate the radius of convergence and the interval of convergence (with examination of the endpoints) for the following power series:
$\sum_{n=0}^{\infty} \frac{x^{2n}]{2n+1}$

Question

Calculate the radius of convergence and the interval of convergence (with examination of the endpoints) for the following power series:
$\sum_{n=0}^{\infty} \frac{x^{2n}]{2n+1}$

anonymous · Answer

$$\sum_{n=0}^{\infty} \frac{x^{2n}}{2n+1}$$

anonymous · Answer

ratio test should do it for this one

anonymous · Answer

$$\lim_{n\to \infty}\frac{2n+3}{2n+1}=1$$ pretty  much by your eyeballs so 
$$|x^2|<1$$ and now check at $x=-1$ and $x=1$

anonymous · Answer

thank you satellite

anonymous · Answer

yw
we still need to check the endpoints

anonymous · Answer

actually there is very little to check, because at both $x=1$ and $x=-1$ you get 
$$\sum\frac{1}{2n+1}$$ which certainly does not converge

anonymous · Answer

yes i looked again it looks good but difficult to figure out how we came to result

anonymous · Answer

we can do it more precisely if you like

anonymous · Answer

yes i would be thankfull for it

anonymous · Answer

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