The problem involves the sequence defined by a_n = 1 + 1/(1*2) + 1/(2*3) + ... + 1/(n*[n+1]) and requires proving that this sequence is bounded above. The subject area relates to series convergence and properties of sequences.
Several participants have provided insights into the structure of the series and its terms. There is a suggestion that the sequence converges to a limit, and one participant questions whether proving the limit by induction would be excessive. The discussion reflects a mix of interpretations and approaches without reaching a definitive conclusion.
Some participants express uncertainty about the completeness of their proofs and seek validation of their reasoning. There is also mention of comparing terms to the reciprocals of squares as a potential bounding argument.
csc2iffy said:ok so this is what i have so far, I'm not sure if it "proves" it completely though:
a1 = 1/1 - 1/2 = 1/(1*2)
a2 = 1/2 - 1/3 = 1/(2*3)
a3 = 1/3 - 1/4 = 1/(3*4)
...
a(n-1) = 1/(n-1) - 1/n = 1/((n-1)*n)
an = 1/n - 1/(n+1) = 1/(n*(n+1))
Adding everything together
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/(n-1) - 1/n) + (1/n - 1/(n+1))
1 - 1/(n+1) = 1/(1*2) + 1/(2*3) + ... + 1/(n*(n+1)) = an - 1
Add 1 to both sides
an = 2 - 1/(n+1) = (2n+1)/(n+1)
lim(an) = 2
Therefore the series is bounded above by 2
does this complete my proof?