Saturday, 15 December 2018

sequences and series - Show that suminftyn=0dfracn2n+1=1



My Work



I felt the best way to go about this problem was to compare it to a well known MacLaurin series. I noticed it resembled the reciprocal of the absolute value of the MacLaurin series of ln(1+x) where x=12 but I had the problem of the n in the numerator still. None of the well known series have an n in the numerator.



The Well Known MacLaurin Series




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My Question



Can someone give me a hint as to which well known MacLaurin series this resembles? Once I have that I know the solution!


Answer



If you want to directly proceed via MacLaurin series, try the MacLaurin series of x2(1x)2, which unfortunately is not there in your list. Else, proceed as follows:
Let Sm=mn=1n2n+1.
Sm=14+28+316+432++m12m+m2m+1Sm2=18+216+332++m22m+m12m+1+m2m+2
now gives us
Sm2=14+18++12m+1m2m+2
I trust you can conclude from this making use of the first MacLaurin series.


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