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Grandi Sum
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Apr 06, 2020
1-1+1-1+1...=1/2 ? Take a look at what I showed. What is going on? I think this is cool and want to share.
1-1+1-1+1...=1/2 ? Take a look at what I showed. What is going on? I think this is cool and want to share.
2 answers2 replies
0
sum can be 0, 1 or 1/2 depending on the method by which you tyr to solve the question. although it is upto infinity so nothing could be said exactly.
There is only ONE method to assign a finite value (if it exits) to an infinite sum and that method is really a definition.
If the infinite sequence of partial sums converges to a finite value then we assign that finite value to the infinite sum. If the infinite sequence of partial sums diverges then the infinite series diverges. No other method is correct or even makes any sense.
s1 = 1
s2 = 0
s3 = 1
s4 = 0
etc ...
Clearly this sequence does not converge. Therefore the original sum does not converge to any finite value and is therefore divergent.
Hello Ian,
You are right. It is exactly what the Wikipedia article and Mathologer explain.
Anyway, there is something noticeable with Grandi's partial sums. They converges on average to 1/2. It is just another way to apprehend it as far as it satisfies some well defined conditions.
Ciao
Hi des,
By using Cesàro summation you can get:
1-1+1-1+...=1/2
Wikipedia:
https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation
Mathologer at 18:28:
https://www.youtube.com/watch?v=jcKRGpMiVTw
This video explains a lot of things dealing with infinite sums.
Bye