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I am trying to solve exercises in my last calculus assignment, which is the following: Give sequences $X_n$, $Y_n$ such that:

1) $X_n$ is not convergent but lim($X_n + Y_n) $ exists.

I did: $X_n= n$, $Y_n$=-n and $(X_n+ Y_n)=0$ $\forall n $

Hence lim$(X_n+ Y_n)=0$.

But my question is: is it possible to give a convergent sequence $Y_n$ such that $X_n$ is not convergent and $(X_n+ Y_n)$ converges?

2) $X_n$ is not convergent but lim($X_nY_n) $ exists.

In this case,I did: $X_n= n$, $Y_n$=1/n and $(X_n+ Y_n)=1$ $\forall n $

My question about thi case is is it possible to give a divergent sequence $Y_n$ such that $X_n$ is not convergent and $(X_nY_n)$ converges?

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    $\begingroup$ 1. If $\lim X_n+Y_n$ exists, and $\lim Y_n$ exists, then their difference $\lim (X_n+Y_n)-Y_n$ also exists. $\endgroup$ – vadim123 Oct 21 '15 at 3:17
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For your first question, no. If $Y_n$ converges and $X_n+Y_n$ converges, then since the difference of two convergent sequences is itself convergent, we have $(X_n+Y_n)-Y_n = X_n$ converges.

For your second question, take $X_n=Y_n=(-1)^n$. This is a divergent sequence. However, $X_nY_n=(-1)^n(-1)^n=1$ which certainly converges. :)

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