Geometric progression

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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:

<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.
How to calculate this? Now, if we look at the n-th element of this sequence, we see:

• <math>(p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) </math>
• <math>=p^1 + p^2 + p^3 + ... + p^{n+1}</math><math> - p^0 - p^1 - p^2 - ... - p^n</math>
• <math>=p^{n+1}-p^0</math>
• <math>=p^{n+1}-1</math>
• <math>\Leftrightarrow</math>
• <math>p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}</math>

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>):

<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 <math>\frac{1}{1-p}</math>.

See also

• Arithmetic progression