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If we build the sum of two power series , $\sum a_kz^k$, $\sum b_kz^k$ both with radius of convergence 1, why does the resulting power series: $\sum (a_k+b_k)z^k$ also have RoC 1?

Is it because : $|\frac{a_k}{a_{k+1}}| = 1 , |\frac{b_k}{b_{k+1}}|=1$

$|\frac{a_k+b_k}{a_{k+1}+b_{k+1}}|$

that doesnt really do anything T_T .... If you see how to show this, please do tell.

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    Not true. The series has radius of convergence *at least* 1, but it may very well be larger.2012-09-13

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The radius of convergence need not stay the same.

Let $p(z)$ have radius of convergence $1$, and let $q(z)$ have radius of convergence $4$. Then $(q+p)(z)$ and $(q-p)(z)$ have radius of convergence $1$, but their sum has radius of convergence $4$.

For any $k\ge 1$, including "$k=\infty$", we can find power series with radius of convergence $1$ whose sum has radius of convergence $k$.

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    Thanks. +++++++2012-09-13