How do I compute the radii of convergence of the power series Σanx^n
(from n=0 to infinity)
with coefficients an=n and an=1/n
Use the ratio test for the power series. With $\displaystyle a_k=k$ thus, $\displaystyle \lim_{k\rightarrow \infty}|\frac{k+1}{k}|=1$ Thus, the radius of convergence is the the reciprocal of that thus $\displaystyle 1/1=1$. However, the interval of convergence is $\displaystyle -1<x<1$. For $\displaystyle x=-1,1$ this power series diverges.Originally Posted by TexasGirl
Now for the second problem, $\displaystyle a_k=1/k, k>1$ use the ratio test for power series again to get $\displaystyle \lim_{k\rightarrow \infty}|\frac{k}{k+1}|=1$ Thus, the radius of converges is 1, thus for $\displaystyle -1<x<1$ converges absolutely. Checking the endpoint (because the ratio test is inconclusive for when its limit is one), we have
$\displaystyle \sum^\infty_{k=1} (-1)^k 1/k$ but this is the alternating-harmonic series thus it converges. For the second possibility we have that $\displaystyle \sum^\infty_{k=1} 1/k$ but this is the harmonic series thus it diverges. Thus, the interval of convergence for the second power series is $\displaystyle -1\leq x<1$
Building from the same power series, if I take q, a nonzero element of C, and put it into the series so that I have the sum of an(q^n)(x^n), can I still use the ratio test in order to find the radius of convergence? Is an the coefficient?
Here is the question I have to answer:
Assume that the power series ∑an(x^n), where n=0 to infinity, has radius of convergence p, where p is a nonnegative real number or stands for the symbol ∞. Let q be an element of C, q≠0. Compute the radius of convergence of ∑an(q^n)(x^n) (n=0 to infinity).
As I understand it, its radius of convergence would be $\displaystyle p/q$ Explanation:
If $\displaystyle \sum^{\infty}_{k=1}a_kx^k$ has radius of convergence of $\displaystyle p$ then by the ratio test for power series $\displaystyle \lim_{k\rightarrow \infty}|\frac{a_{k+1}}{a_k}|=1/p$ because as I said it is the reciprocal of the limit. Thus, the new infinite series given by $\displaystyle \sum^{\infty}_{k=1}a_kq^kx^k$ then by the ratio test $\displaystyle \lim_{k\rightarrow \infty}|\frac{a_{k+1}q^{k+1}}{a_kq^k}|=\frac{q}{p}$ thus, the radius of convergent is the reciprocal thus $\displaystyle \frac{p}{q}$