Question

give an example of two divergent sequences X and Y such that a. X + Y converges b. XY converges

Answer 1

I have no clue im more of a
History person

Answer 2

Let \(X_n=(-1)^n\) and \(Y_n=(-1)^{n+1}\). Both clearly are divergent sequences.

For even \(n\), \(X_n=1\) and \(Y_n=-1\), so \(X_n+Y_n=1+(-1)=0\).

For odd \(n\), \(X_n=-1\) and \(Y_n=1\), so again, \(X_n+Y_n=0\).

You can use the same sequences \(X\) and \(Y\) for the second question, since
\[X_nY_n=(-1)^n(-1)^{n+1}=(-1)^{2n+1}=-1\]