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Do the following have same radius of convergence?

$1. \sum^\infty_{n=0} any\ power\ series$ and $\sum^\infty_{n=1} the \ same\ power\ series$

Suppose $\sum a_nx_n$ has radius of convergence $R$

$2. \sum^\infty_{n=1} na_nx^{n-1}$ and $\sum^\infty_{n=1} na_nx^{n}$

$3. \sum^\infty_{n=0} \frac{a_n}{n+1}x^{n+1}$ and $\sum^\infty_{n=0} \frac{a_n}{n+1}x^{n}$

For $1$, I couldn't think of a counterexample

For each of $2$ and $3$, on a textbook it says that they are differ in ratio by a constant $x$ so the radius of convergence is the same, but I don't quite get the meaning? So personally I used general series ratio test. So for $2_1$, I have $R'=\lim\frac{(n)a_nx^{n-1}}{(n+1)a_{n+1}x^{n}}=R$ (since $\sum a_nx_n$ has radius of convergence $R$), for $2_2$, I have $R'=\lim\frac{(n)a_nx^{n}}{(n+1)a_{n+1}x^{n+1}}=R$. So they are same. Same procedure for $3$.

Is it correct?

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  • $\begingroup$ For 1, finite number of terms do not change the nature of the series. $\endgroup$ – Sahiba Arora Jun 16 '17 at 13:06
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You just need to know that

by ratio test

$\sum a_nx^{n-1} $ and $\sum a_n x^n $ have the same radius .

$\sum a_nx^n $ and $\sum na_nx^n $ have the same radius.

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