Notation to express $(n^1+n^2+...+n^k)$.

Question

What are some mathematical conventions for expressing $(n^1+n^2+...+n^k)$ in a simpler format?

Answer 1

This is a geometric series. You could write it as $\sum \limits_{j=1}^k n^j$

If $S = n^1+n^2+n^3+n^4 +\cdots +n^k$

then $nS=n^2+n^3+n^4 +\cdots +n^k+n^{k+1}$

so by subtraction $(n-1)S= n^{k+1} -n^1$

and thus $S = \dfrac{n}{n-1}(n^k-1)$

Answer 2

There is a sum-notation:

$$\sum_{i=1}^k n^i=n+n^2+\dotso +n^{k-1}+n^k$$

Also this is a geometric sum so there is an explicit formula for it given by:

$$\frac{n(n^k-1)}{n-1}$$