Hey everyone, these are 3 of like 15 problems that I need help with. I don't expect someone to do all three, but to pick one they are confident with and try to help me. If you can, thanks =D
1)
Evaluate (Find the sum of):
Thanks
Hey everyone, these are 3 of like 15 problems that I need help with. I don't expect someone to do all three, but to pick one they are confident with and try to help me. If you can, thanks =D
1)
Evaluate (Find the sum of):
Thanks
Can you simplify this sum?
$\displaystyle \sum\limits_{k = 0}^3 {\ln \left( {\frac{{k + 1}}{{k + 3}}} \right)} = \left( {\ln (1) - \ln (3)} \right) + \left( {\ln (2) - \ln (4)} \right) + \left( {\ln (3) - \ln (5)} \right) + \left( {\ln (4) - \ln (6)} \right) = ?$
If so look for the pattern.
Hello, PapaSmurf!
$\displaystyle \text{Evaluate: }\:S \;=\;\sum^{125}_{k=0} \ln\left(\frac{k+1}{k+3}\right)$
We have: .$\displaystyle S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128}) $
. . $\displaystyle S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right) $
. . . . $\displaystyle =\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right) $
. . . . $\displaystyle =\;-\ln8128 \;=\;-9.00307017$