Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$.
Intuitively this doesn't make sense to me, If you have, $$\sum_{n=n_{0}}^{} (a_{n}+b_{n})x^n$$ Shouldn't this be equal to ;
$$\sum_{n=n_{0}}^{} (a_{n}+b_{n})x^n=\sum_{n=n_{0}}^{} (a_{n})x^n+\sum_{n=n_{0}}^{} (b_{n})x^n$$
And intuitively …at least for me, shouldn't the radius of convergence of the LHS be the minimum of the radii of convergence of the two power series on the RHS?
I can't think of a case for when it is larger than the minimum of the radii of convergence of the two power series on the RHS.
Any help would be much appreciated.
Best Answer
You are correct: as long as both series converge (that is, as long as $\lvert x\rvert\leq\min\{R,S\}$, so that you are inside both radii of convergence), you have $$ \sum_{n=n_0}^{\infty}(a_n+b_n)x^n=\sum_{n=n_0}^{\infty}a_nx^n+\sum_{n=n_0}^{\infty}b_nx^n. $$ But, this doesn't mean that $\min\{R,S\}$ is the best that we can do!
For an example of that, think about the case where there is a lot of cancellation of terms in $a_n+b_n$. Do you see what I'm getting at?