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What is the value of the following sum?
S = 12/21 + 22/22 + 32/23 + … + n2/2n + …
As usual, watch the video for a solution.
Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Answer To Sum of n squared over 2 to n
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
First we will check for convergence.
Now let’s solve it in a few ways.
Method 1: pattern
S = 12/21 + 22/22 + 32/23 + … + n2/2n + …
S = 1/2 + 4/4 + 9/8 + 16/16 + 25/32 + …
2S = 1 + 4/2 + 9/4 + 16/8 + 25/16 + …
2S – S = S = 1 + 3/2 + 5/4 + 7/8 + 9/16 + …
S – 1 = 3/2 + 5/4 + 7/8 + 9/16 + …
2(S – 1) = 3 + 5/2 + 7/4 + 9/8 + …
2(S – 1) – S = 2 + 2/2 + 2/4 + 2/8 + …
Simplifying both sides we get:
S – 2 = 2 + 1 + 1/2 + 1/4 + …
S – 2 = 2 + 1/(1-1/2)
S – 2 = 2 + 2
So we have:
S = 6
Method 2: sequences
The sequence n/2n is an arithmetico-geometric series and the series 1/2n is a standard geometric series.
Method 3: generating function
References
Answers by Trevor, Aryan Arora, Alexey Godin
https://www.quora.com/How-do-you-evaluate-the-sum-of-n-2-2-n-from-n-1-to-infinity
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