if I have $$ \sum_{n=0}^{\infty}\sum_{p=0}^{n} (\cos)^{n-p} (i\sin)^{p} x^n $$ Now substituting $x^n= x^{n-p}x^{p}$ and using the transformation identity $$\sum_{j=1}^{\infty}\sum_{i=1}^{j}f(i,j)=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}f(i,j)$$ Here i = p; j = n-p Then $$=\sum_{p=0}^{\infty}\sum_{n-p=0}^{\infty} (\cos)^{n-p} (i\sin)^{p}x^{n-p}x^{p} $$ Can I group it like $$ =\sum_{n-p=0}^{\infty}(\cos)^{n-p}x^{n-p}\sum_{p=0}^{\infty}(i\sin)^{p}x^{p} $$ Is my final equation correct?
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