## anonymous 4 years ago Help needed... sum of series = 1 + (3/4) + (7/16) + (15/64) + (31/256) + .. infinity

1. anonymous

firstly is it a g.p?

2. anonymous

okk I got this much that = 1 + (2^2 -1)/2^2 + (2^3 -1)/(2^4) + (2^4 - 1)/2^16 + ... = $2^2 * \sum_{0}^{\infty} (2^n-1)/(2^2)^n$ Now what ????

3. anonymous

its not a gp.. its a combination .. think someone with little practice can answer this .. I have lost touch and hence have forgot the method to sum these..

4. anonymous

also .. my formula is wrong .. I think it would be somewhat like (2^(n+1) - 1)/(2^(2(2n-2))

5. anonymous

$\frac{2^{n} -1}{2^{2n}} = \frac{1}{2^{n}}-\frac{1}{2^{2n}} =(\frac{1}{2})^{n} - (\frac{1}{4})^{n}$ $\rightarrow 4*[\sum_{1}^{\infty}(\frac{1}{2})^{n} - \sum_{1}^{\infty}(\frac{1}{4})^{n}]$

6. anonymous

no i think your formula worked if you start at n=1

7. anonymous

@dumbcow .. no I mistyped the formula .. look at the series ... you won't get 7/16 .. or 31/256 ... it is a little more complicated...

8. anonymous

(2^5 -1)/(2^2)^4

9. anonymous

got it ... thanks .. yeah my formula is just slightly of.. but thanks anyways.. :)

10. anonymous

$$\large \frac 23$$ isn't ?

11. anonymous

it cannot be 2/3 because it is (1 + 3/4 + ... ) and all terms are positive

12. anonymous

no thats right for the summation part. multiply by 4 and you get , sum = 8/3

13. anonymous

yeah ... it wd be 8/3 ..

14. anonymous

thanks both of you.. :)

15. anonymous

yes it should be 8/3, I summed up dumcow's generalization, but the actualy generalization should be $$\huge \sum\limits_{k=0}^\infty \frac {2^{k+1} -1}{2^{2k}}=\frac 83$$

16. anonymous