Assume we have
a1+a2+a3+...+an+...
and
b1+b2+b3+...+bn+...
Assume that these are two conditionally convergent series whose sums are FIRST and SECOND.
Consider the infinite series obtained by writing the bi after the ai:
a1+a2+a3+...+an+...+b1+b2+b3+...+bn+..…

Questions:
Is this series convergent?
If affirmative, will its sum be FIRST+SECOND?
Please give a detailed proof to support your answer.

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