Show that the power series $\sum_{n=0}^{\infty}{c_{n}x^{n}}$ has the same radius of convergence as $\sum_{n=0}^{\infty}{c_{n+m}x^{n}}$ for any positive integer $m$.
Should I use the Ratio Test? Or do I use induction?
[Math] Show that two power series have the same radius of convergence
convergence-divergencepower seriesreal-analysis
Best Answer
The radius of convergence $R$ can be characterized as follows:
Let's put aside the cases $R=0$ and $R=\infty$, for the moment.
There is no problem with convergence at $0$, so we can assume $x\ne0$.
Suppose $|x|<R$. Then the series $\sum\limits_{n\ge0}c_nx^n$ converges, so also $$ \sum_{n\ge1}c_nx^n $$ does and, by multiplying by $1/x$, also $$ \frac{1}{x}\sum_{n\ge1}c_nx^n=\sum_{n\ge1}c_nx^{n-1}= \sum_{n\ge0}c_{n+1}x^n $$ converges.
Conversely, if $\sum\limits_{n\ge0}c_{n+1}x^n$ converges, then also $$ x\sum_{n\ge0}c_{n+1}x^n=\sum_{n\ge0}c_{n+1}x^{n+1}=\sum_{n\ge1}c_{n}x^n $$ converges and so does $\sum\limits_{n\ge0}c_nx^n$.
Now you can start induction.
For the case $R=\infty$ it's the same, because it means that the series $\sum\limits_{n\ge0}c_nx^n$ converges for all $x$.
Do separately the case $R=0$.