Monday, March 4, 2013

Infinite Series

Robert had trouble understanding how it is possible for an infinte series to converge. I asked him to consider an exponential expression with a base that is a fraction less than one. First I asked him to consider 1/2 squared. After he found 1/4, I asked him to consider whether this number is larger or smaller than 1/2. He wrestled with the concept for a moment, and then realized that 1/2 times 1/2 is the same as a half of 1/2, which is of course a smaller number.
I then asked him to consider 1/2 cubed. He was quickly able to understand that this number is smaller yet. I then asked him to consider 1/2 raised to the 1000th power. This number is very close to zero.
Robert was then able to understand that as x approaches infinity, (1/2)^x will approach 0. This enabled him to understand how an infinite series can converge.

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