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I came across the following two series while trying to solve Laplace's equation in two dimensions.

$$T_1 = \sum_{n=1}^{\infty}\rho^n(A_n\cos(n\phi) + B_n \sin(n\phi))$$

$$T_2 = \sum_{n=1}^{\infty}\rho^{-n}(C_n \cos(n\phi) + D_n \sin(n\phi))$$

The coefficients $A_n$, $B_n$, $C_n$, $D_n$ are to be determined from suitable boundary conditions. The domain over which these sequences are defined are $0<=\rho<\infty$ and $0<=\phi<=2\pi$ Now, my questions are: 1) Is it possible for $T_1$ to converge as $\rho\rightarrow\infty$ for a suitable non trivial (not all zero) choice of coefficients $A_n$ and $B_n$? 2) Is it possible for $T_2$ to converge as $\rho\rightarrow 0$ for a suitable non trivial (not all zero) choice of coefficients $C_n$ and $D_n$? 3) Is it possible for these sequences to converge at the above mentioned points even though the individual terms of the sequence blow up. 4) Is it possible to uniquely determine the coefficents $A_n$, etc., simply based on the fact that these two sequences converge? Would appreciate any insight on this. Thanks.

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