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Series
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Main Concept
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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 series can be written as follows:
![(∑)x[k ]=x[1]+x[2]+x[3]+...+ x[k]](/support/helpjp/helpview.aspx?si=8367/file02333/math42.png) , where ![x[k] in `ℝ` for every k =1... n](/support/helpjp/helpview.aspx?si=8367/file02333/math44.png) .
An infinite series series can be written as follows:
![(∑)x[k ]=x[1]+x[2]+x[3]+...](/support/helpjp/helpview.aspx?si=8367/file02333/math55.png) , where ![x[k] in `ℝ` for every k in `ℕ`](/support/helpjp/helpview.aspx?si=8367/file02333/math57.png) .
Sometimes the terms of an infinite series can be added up to give a finite number, called the sum of the series.
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Consider the series  . Does this series have a sum?
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Answer
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The sum of the series is 1.
To see this, imagine that we paint a blank canvas in steps. At each step, you paint half of the unpainted area. The total area painted after  steps is therefore the  th partial sum,
![(∑)(1/(2^(k)))[ ]=1/(2)+1/(4)+1/(8)+1/(16)+1/(2^(n))](/support/helpjp/helpview.aspx?si=8367/file02333/math124.png)
. The total area remaining unpainted is  . 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 press the Paint All button.
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