I'm trying to solve a problem on convolution from Alan V.Oppenheim:
Find the convolution output $y[n]$ for the following signals:
$x[n]= u[n]$ and $h[n]=a^{n}u[-n-1], a>1 $
I started the evaluation:
$$y[n]=\sum_{k=-\infty}^{+\infty} u[k]a^{n-k}u[-n+k-1]$$ considering that $u[k]=1$ for $k>=0$ and $u[-n+k-1]=1$ for $k>=n+1$ which I evaluated to $$y[n]=a^{n}\sum_{k=m}^{+\infty} a^k$$ where $m=n+1$ could be $<0$ or $>0$ and I tried to evaluate for $m>0$ which is the same as $n>-1, which evaluated as:
$$y[n]=a^{n}\sum_{k=m}^{+\infty} a^m=a^{n}((\sum_{k=0}^{+\infty} a^m)-(\sum_{k=0}^{m-1} a^m))=a^{n}((\frac{1}{1-a^{-1}})-(\frac{1-a^{-m}}{1-a^{-1}}))=\frac{a^{n-m}}{1-a^{-1}}=\frac{a^{-1}}{1-a^{-1}}$$ since n-m=1,
but when I evaluated for m<0 which is n<=-1 I'am facing a problem:
$$y[n]= a^{n}((\sum_{k=m}^{-1} a^m)+(\sum_{k=0}^{+\infty} a^m))$$
How do I evaluate the first summation. I mean am I to consider k=-m since m<0?