Geometric Series
Main Concept
A series is the sum of terms in a sequence. There are two types of series: finite and infinite. Finite series have defined first and last terms, and infinite series continue indefinitely.
A finite series can be written as follows:
∑k =1nxk =x1+x2+x3+...+ xk , where xk∈ℝ for every k=1..n
An infinite series can be written as follows:
∑k =1∞xk =x1+x2+x3+... , where xk∈ℝ for every k∈ℕ
A geometric series is a series in which the term xk+1 can be obtained from the previous term xk by multiplying by a fixed number. For instance, ∑n=1∞12n is a geometric series in which each successive term is found by multiplying the previous term by 12.
Sometimes the terms of an infinite series can be added up to give a finite number, called the sum of the series.
Consider the geometric series ∑n =1∞12n. Does this series have a sum?
Answer
∑k =1∞12k=12+14+18+116+ ... = 1
The sum of the series is 1.
To see this, imagine that we paint a blank canvas in steps. At each step, we paint half of the unpainted area. The total area painted after n steps is therefore the nth partial sum, ∑k =1n12k =12+14+18+116+12n. The total area remaining unpainted is 12n. After an infinite number of steps we will have painted all of the canvas, of which the area is 1.
Click on the canvas to paint one section, or click "Paint All".
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