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# Something I realized about powers:

I was walking around (currently on vacation) and I realized something which I think is true (and I'm sure have been thought of before):

For all positive integer $$b > 1$$...

$$\sum_{n = 0}^k b^n = \frac {b^{n+1} - 1}{b - 1}$$

So I ask a few questions:

1. Can somebody prove this? I'm only a high school student so I'm not totally sure how to do this.

2. Is this true for non-integers?

3. Any relevant theorems on this?

Thanks guys

Note by Michael Tong
3 years, 7 months ago

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Is not this, the sum of a geometric progression? I am sure you already know how to get this formula. · 3 years, 7 months ago

The series is nothing but a G.P. Take the sum as S and multiply S by b, now subtract S from bS and you will get the result. Simple!!! · 3 years, 7 months ago

Umm gp? · 3 years, 7 months ago

S = 1 + b + b^2 + b^3 +....+ b^n

Multiply b both sides. We have two equations -

S = 1 + b + b^2 + b^3 +....+ b^n

b*S = b + b^2 + b^3 + .....+ b^n+1

Subtract the two equations. You will get -

S(b - 1) = b^n+1 - 1

Multiply (b - 1) both sides and there you go! · 3 years, 7 months ago

One more, since this proof is valid for every real number except 1 I suppose, this theorem works for all real numbers except 1. I am not 100% sure though. Please tell me if there are some other numbers other than 1. · 3 years, 7 months ago

Theorem: The series $$\sum a_n$$ converges if $\lim_{n\to \infty} \sup \sqrt[n]{}{\left |a_n \right |<1.}$

Definition:

http://en.wikipedia.org/wiki/Supremum

Let $${s_n}$$ be a sequence of real numbers. Let $$E$$ be the set of numbers x such that $$s_{n_k} \to x$$ for some subsequence $${s_{n_k}}$$. Put $\lim \sup x_n= \sup E.$

Proof of the Theorem:

If $\alpha=\lim_{n\to \infty} \sup \sqrt[n]{\left |a_n \right |}$ then there exists $$\beta >0$$ such that $$\alpha< \beta<1$$. Then there there exists $$N \geq 0$$ such that $$N \leq n$$ implies $\left |a_n \right |\leq \beta ^n$. Then the convergence of geometric series implies the convergence of this series here.

From this theorem, you can evaluate 'radius of convergence' for a complex power series. · 3 years, 7 months ago