How can one simplify 36+3⋅56⋅9+3⋅5⋅76⋅9⋅12 up to ∞?
I am new to the topic so can the community please guide me on the approach one needs to take while attempting such questions?
Things I am aware of:
Permutations and combinations
Factorials and some basic properties that revolve around it.
Some basic results of AP, GP, HP
Basics of summation.
Thanks for reading.
Answer
Let S=36+3⋅56⋅9+3⋅5⋅76⋅9⋅12⋯⋯∞
Then S3=1⋅33⋅6+1⋅3⋅53⋅6⋅9+1⋅3⋅5⋅73⋅6⋅9⋅12+⋯⋯
So 1+13+S3=1+13+1⋅33⋅6+1⋅3⋅53⋅6⋅9+1⋅3⋅5⋅73⋅6⋅9⋅12+⋯⋯
Now campare the right side series
Using Binomial expansion of (1−x)−n=1+nx+n(n+1)2x2+n(n+1)(n+2)6x3+.......
So we get nx=13
We get 13(13+x)=13⇒x=23
So we get n=12
So our series sum is (1−x)−n=(1−23)−12=√3
So 43+S3=√3
So S=3√3−4.
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