can anybody show me or give me a hint on how the following two series are equal?
I have been trying to figure out the simplification of the first series to the second. i have been trying to apply exponential rule however i couldn't figure out how. I think i have missed something related to the starting point of the series.
$$\sum_{i=k+1}^{\infty} ix^{i-k}$$
and
$$\sum_{j=1}^{\infty}(k+j)x^j$$
sorry for the format of the series.
On
This is called shifting indices.
Just look at the first few terms: $$\sum_{i=k+1}^{\infty} ix^{i-k}=(k+1)x^{1} +(k+2)x^{2}+(k+3)x^{3}+...... $$ $$\sum_{j=1}^{\infty}(k+j)x^j = (k+1)x^{1} +(k+2)x^{2}+(k+3)x^{3}+...... $$ They are equal term by term.
You subtracted $k$ from the bounds but you added the same $k$ to the index within the sum, so you get the same terms.
You can substitute $j=i-k$. In that case, as $i$ goes from $k+1$ to $\infty$, $j$ will go from $1$ to $\infty$, and $i=k+j$:
$$\sum_{i=k+1}^{\infty} ix^{i-k}=\sum_{j=1}^{\infty}(k+j)x^j$$
The substitution here is no different from any other substitution in algebra. The two series have the same terms:
$$(k+1)x+(k+2)x^2+(k+3)x^3+\cdots$$
just written using $i$ in the first sum, and $j$ in the second. Therefore, they are the same.