<s>[INST] Summation of series and Taylor series are giving different results $A = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots$ 
$B = 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots$ 
$A + B = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots$ 
$2A = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots$ 
$2A = A + B$
$A - B = 0$
$0 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots$  
Using Taylor expansion for $\ln(1+x)$
$\ln(2) = 1 -\frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots$
Where did I go wrong ?
 [/INST] What you did wrong was to assume that you can deal with divergent series as if they were convergent. If you say that
$$
A=\frac12+\frac14+\frac16+\cdots
$$
then $A$ is a name of the series, but it is not a number (since the series diverges).
Therefore
$$
1+\frac12+\frac13+\frac14+\cdots
$$
is just another (divergent) series and it is an unfortunate option to call it $2A$.
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