Formulas & tables/Mathematics/Sequences/Sum of a convergent geometric series

Sum of a convergent geometric series

S=a11−qS=\dfrac{a_1}{1-q}

Result unit: — (liczba)

What it describes

The sum of an infinite convergent geometric series (when ∣q∣<1|q|<1).

Symbols

SS
sum of the series
a1a_1
first term
qq
common ratio (∣q∣<1|q|<1)

Where it comes from

When ∣q∣<1|q|<1, the successive terms decrease so fast that the infinite sum is finite. It is the limit of the partial sums.

When to use it

Infinite geometric sums (repeating decimals, 0.999…=10.999\ldots=1, Zeno's paradox).

Example

For a1=1a_1=1, q=12q=\tfrac12 compute the sum of the series.
Substitutiona1=1a_1=1, q=0.5q=0.5.
ResultS=11−0.5=2S=\dfrac{1}{1-0.5}=2.

Related formulas

Sum of a geometric sequence Geometric sequence — nth term

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