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Challenge: Explain the seemingly contradictory results. (i.e. find the error)?
Sum (n = 1 to infinity) of
ln [ n / (n+1) ]
When summing explicitly as written, we get:
ln(1/2) + ln(2/3) + ln(3/4) + ln(4/5) + ... we can see that the upper bound of the sum is ln(1/2) and that the sum should become more negative as we add smaller and smaller terms.
However, let's write it another way:
ln [ n / (n+1) ] = ln n - ln(n+1)
So we get:
[ln1 - ln2] + [ln2 - ln3] + [ln3 - ln4] + [ln4 - ln5] + ....
Use associative property of addition:
ln1 + (-ln2 + ln2) + (-ln3 + ln3) + (-ln4 + ln4) + (-ln5 + ln5) + ...
= ln 1 + 0 + 0 + 0 + 0 + 0 + ....
= ln1
= 0
1 Answer
- Anonymous9 years agoFavorite Answer
I can specifically tell you what is wrong with the second part:
ln 1 + (-ln 2 + ln 2)... ln n - ln(n + 1)
(ln n cancels out with each inside term, so we can camel it out but we know the last term is ln(n + 1), so we leave it)
= ln 1 - ln (n + 1)
ln 1 = 0 - ln (n + 1)
lim --> ∞ -ln(n + 1) = -∞, not 0
The series is divergent. That is what is wrong with it.