Thursday, 2 April 2015

sequences and series - How can one simplify frac36+frac3cdot56cdot9+frac3cdot5cdot76cdot9cdot12 up to infty?




How can one simplify 36+3569+3576912 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:




  1. Permutations and combinations


  2. Factorials and some basic properties that revolve around it.


  3. Some basic results of AP, GP, HP



  4. Basics of summation.




Thanks for reading.


Answer



Let S=36+3569+3576912



Then S3=1336+135369+135736912+



So 1+13+S3=1+13+1336+135369+135736912+




Now campare the right side series



Using Binomial expansion of (1x)n=1+nx+n(n+1)2x2+n(n+1)(n+2)6x3+.......



So we get nx=13

and nx(nx+x)2=1336



We get 13(13+x)=13x=23



So we get n=12




So our series sum is (1x)n=(123)12=3



So 43+S3=3



So S=334.


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