How to prove this.

1. Dec 2, 2005

viren_t2005

how to prove that 1/2+1/4+1/8+............+1/2^n is always less than one?

2. Dec 2, 2005

benorin

The partial sum of the geometric series is:

$$a+ar+ar^2+\cdot\cdot\cdot +ar^{n}=a\frac{1-r^{n+1}}{1-r}$$

3. Dec 2, 2005

CRGreathouse

Tongue-in-cheek

1. Suppose the partial sum $$S_n$$ > 1.
2. We know $$S_n<\lim_{n\rightarrow\infty}S_n=1$$.
3. So $$1<S_n<1$$, a contradiction.

4. Dec 4, 2005

Werg22

$$S = a+ar+ar^2+\cdot\cdot\cdot +ar^{n}$$

$$rS = ar+ar^2 ar^3 +\cdot\cdot\cdot +ar^{n} + ar^{n+1}$$

$$rS - S = ar^{n+1} - a$$

$$S = a\frac{r^{n+1} - 1}{r-1}$$

For the general case it's

$$ar^{c} + \cdot\cdot\cdot +ar^{n}=a\frac{r^{n+1} - r^{c}}{r-1}$$

Where n>c

c is lowest power of the sum and n the highest.

In your case, the formula would be

$$\frac{2^{-1+1} - 2^{n}}{1}$$

$$1 - 2^{n}$$

2^n is always a positive integrer, thus the sum is alwasy inferior to 1.

Last edited: Dec 4, 2005
5. Dec 4, 2005

Tide

You could do a geometric proof. Start with a unit square. Cut it into two equal parts and remove one of them. Repeat the process on the remaining half and so on. Your series represents the sum of the areas of the pieces you have removed and can never exceed 1 with a finite number of repetitions since some portion of the original square remains.