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==geometric series== just a start
::2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>
::2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>
This is why geometric progressions are sometimes called "exponential growth."
This is why geometric progressions are sometimes called "exponential growth."
==Geometric Series==
Often, the sequence of partial sums of a geometric progression <math>(p^0, p^1, p^2, p^3,...)</math> is of some interest (vide: we are starting with the exponent zero here.) This sequence would be: <br />
<math>(p^0, p^0+p^1, p^0+p^1+p^2, p^0+p^1+p^2+p^3, ...)</math> and is called a '''Geometric Series'''. <br>How to calculate this? Now, if we look at the n-th element of this sequence, we see: <ul><li><math>(p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) </math><li><math>=p^1 + p^2 + p^3 + ... + p^{n+1}</math><math> - p^0 - p^1 - p^2 - ... - p^n</math><li><math>=p^{n+1}-p^0</math><li><math>=p^{n+1}-1</math><li><math>\Leftrightarrow</math><li><math>p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}</math></ul>Obviously, the last step is allowed only if <math>p \neq 1 </math>. So, the sequence of partial sums is (if <math>p \neq 1 </math>):<br>
<math>\frac{1}{p-1} (p^1-1, p^2-1,p^3-1, ...)</math> - and it will converge for <math>-1 < p < 1 </math> to the [[Limit (mathematics)|limit]] <math>\frac{1}{1-p}</math>.
[[Category:mathematics]]
[[Category:mathematics]]