#$5$ For two sequences $\{a(n){\}}_n=1^{}$ and $\{b(n){\}}_n=1^{},$ define

$\{a(n){\}}_n=1^{}*\{b(n){\}}_n=1^{}=\{{\displaystyle \underset{k=1}{\overset{n}{}}}a(k)b(n-k+1){\}}_n=1^{}$

Show that

$\{1,-1,2,-2,3,-3,4,-4,5,-5$cdots$\}*\{1,2,3,4,5,6,7,8,9,10,$cdots$\}=\{1,1,3,3,6,6,10,10,15,15,$cdots$\}$