Content deleted Content added
No edit summary |
|||
Line 41:
Consider the final transformation that you see in the second animation, and we compare this figure with the square base parallelepiped that contains it. We expect that the ratio between these figures becomes 1/2 to infinity. Performing calculations with Excel, you see that this is true. In addition, as has happened in the discussion at the link http://upload.wikimedia.org/wikipedia/commons/6/6c/Pubblicazione_english.pdf, you encounter these other amazing results:
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>3</sup>)/(Σ<sub>n</sub> n).
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>5</sup>)/(Σ<sub>n</sub> n).
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>7</sup>)/(Σ<sub>n</sub> n).
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>9</sup>)/(Σ<sub>n</sub> n).
<sup>n→∞ </sup><sup>1</sup>
which, by induction, can be generalized in a formula. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers.
| |||