Combining expression to form a single power series

Question

$$\sum_{k=0}^{\infty}x^{k+2}+\sum_{k=0}^{\infty}x^k$$

I've tried to combine the expression into one power series whose general term is $x^k$. Is there a way of combining this without substitution?

Answer 1

Just write out the first few terms:

$$\sum_{k=0}^{\infty} x^{k+2} + \sum_{k=0}^{\infty} x^k \\ =\big(x^2 + x^3 + x^4. . . .\big) + \big(1+x+x^2+x^3+x^4. . . .\big) \\ =1 + x + 2\big(x^2 + x^3 + x^4 . . . .\big) \\ = 1+x +2\sum_{k=0}^{\infty} x^{k+2}$$

I doubt there is way to merge this into a single series.