A function (f) is defined by f(x) = 1 + 2x + x^2 + 2x^3 + x^4 + ...
that is, its coefficients are c2n = 1 and c2n+1 = 2 for all n > or = to 0. Find the interval of convergence of the series and find an explicit formula for f(x).
A function (f) is defined by f(x) = 1 + 2x + x^2 + 2x^3 + x^4 + ...
that is, its coefficients are c2n = 1 and c2n+1 = 2 for all n > or = to 0. Find the interval of convergence of the series and find an explicit formula for f(x).
Normally one would find the radius of convergence of a power series by using the "ratio test"- take the limit, as n goes to infinity of $\displaystyle \frac{|a_{n+1}x^{n+1}|}{|a_nx^n|}$- but here that does not converge because each fraction is either x/2 or 2x.
So use the "root test" instead. If n is even, $\displaystyle |a_n x^n|= |x^n|$ and the nth root is |x|. If n is odd, [tex]|a_n x^n|= |2x^n| and the nth root is [tex]\sqrt[n]{2}|x|. As n goes to infinity, both of those go to |x|. The sequence converges as long as that limit is less than 1 so the radius of convergence is 1.
(But chisigma's answer is really really clever!)