Geometric progression

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Geometric series

Often, the sequence of partial sums of a geometric progression $(p^{0},p^{1},p^{2},p^{3},...)$ is of some interest (vide: we are starting with the exponent zero here.) This sequence would be:

$(p^{0},p^{0}+p^{1},p^{0}+p^{1}+p^{2},p^{0}+p^{1}+p^{2}+p^{3},...)$ and is called a Geometric Series.
How to calculate this? Now, if we look at the n-th element of this sequence, we see:

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

Obviously, the last step is allowed only if $p\neq 1$. So, the sequence of partial sums is (if $p\neq 1$):

${\frac {1}{p-1}}(p^{1}-1,p^{2}-1,p^{3}-1,...)$ - and it will converge for $-1<p<1$ to the limit ${\frac {1}{1-p}}$.

See also

• Arithmetic progression