How to denote $\underbrace{(1+n(1+n(1+\cdots)))}_{x-1 \text{times}}$ in a single expression?

Question

I learnt that, $$ n + n^2 + n^3 + n^4 +. . . + n^x = n\underbrace{(1+n(1+n(1+\cdots)))}_{x-1 \text{times}} $$ So,I want to denote RHS in a single expression.

Ignore the Horner form; simplify the LHS with the geometric sum formula. — anon, Aug 04 '13 at 10:11

T-Wayne · Answer 1 · 2013-08-04T14:24:42.090

4

Multiply the left hand side by $(1 - n)$. This will collapse it down to $n - n^{x+1}$. Then divide by $(1 - n)$ to get a final answer of:

$$S = \frac{n - n^{x+1}}{1 - n}.$$

For more on Telescoping Series

edited Aug 04 '13 at 14:24

answered Aug 04 '13 at 10:14

T-Wayne

123

MJD · Answer 2 · 2013-08-04T15:52:46.593

2

You could say $$\text{Let }f(y) = n\cdot(1+ y)$$

and then the expression on the right becomes $$f^{x-1}(n).$$

edited Aug 04 '13 at 15:52

answered Aug 04 '13 at 14:06

MJD

65,394
39
298
580

Given you suppose that exponent denotes iteration, and not power ;) – yo' Nov 01 '13 at 12:46

How to denote $\underbrace{(1+n(1+n(1+\cdots)))}_{x-1 \text{times}}$ in a single expression?

2 Answers2