Grandi's Series and Thomson's Lamp

In this article, I'm going to talk about a very simple yet perplexing series: Grandi's series. Grandi's series is:
1 - 1 + 1 - 1 + 1 - 1 +...

You could write this series as:

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) +...

and get the value of the series to be 1. Or, you could write the series as:
(1 - 1) + (1 - 1) + (1 - 1) +...

and get a value of 0.

Shocked? Prepare for more! If I label Grandi's series as S, then 1 - S will be:

1 - (1 - 1 + 1 - 1 + 1 - 1 +...)

= 1 - 1 + 1 - 1 + 1 - 1 + 1...

= S

So, we have 1 - S = S, which means that S = 1/2. What? How can 1 - 1 + 1 - 1 + 1... be 1/2? How can the sum of integers possibly be a proper fraction? 
Surprisingly, most mathematicians accept the value of Grandi's series to be 1/2!

Now, let me tell you about another series:


This is a much friendlier series than Grandi's series, because it has a fixed value that we can calculate. Infinite series that have a fixed value are known as convergent series. They are called convergent because they converge to a particular value. Series like Grandi's series (series that don't have a fixed value) are known as divergent series. 

Let's calculate the value of sn. Multiplying sn by 2, we get:


Subtracting sn from both sides, we get:


Thus, as n approaches infinity, sn tends to 1.

Now, I'm gonna twist your minds with this question. Consider a lamp with a switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Let's say you start a timer, and after a minute you turn the lamp on. After half a minute, you turn the lamp off. After a quarter of a minute, you turn the lamp on again. After an eighth of a minute, you turn the lamp off. Keep doing this. The total time will be: 

1 + 1/2 + 1/4 + 1/8 + 1/16 +... = 1 + sn

which has a value of 2 as n approaches infinity. 

You can represent "on" as 1 and "off" as 0. At the end of two minutes, will your lamp be on or off?




Grandi's series generates the sequence of values {1, 0, 1, 0, ...}, which represents the changing state of the lamp. 

According the mathematicians, Grandi's series has a value of 1/2, which means that the lamp is on and off simultaneously. But how can that be possible? 

This is the famous paradox known as Thomson's lamp. 

Thanks for reading, please share your thoughts!



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