How to do a Summation

• May 14th 2011, 09:25 AM
PapaSmurf
How to do a Summation
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):

http://i54.tinypic.com/30cwydu.jpg

Thanks
• May 14th 2011, 09:37 AM
Plato
Can you simplify this sum?
$\sum\limits_{k = 0}^3 {\ln \left( {\frac{{k + 1}}{{k + 3}}} \right)} = \left( {\ln (1) - \ln (3)} \right) + \left( {\ln (2) - \ln (4)} \right) + \left( {\ln (3) - \ln (5)} \right) + \left( {\ln (4) - \ln (6)} \right) = ?$

If so look for the pattern.
• May 14th 2011, 10:06 AM
Soroban
Hello, PapaSmurf!

Quote:

$\text{Evaluate: }\:S \;=\;\sum^{125}_{k=0} \ln\left(\frac{k+1}{k+3}\right)$

We have: . $S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128})$

. . $S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right)$

. . . . $=\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right)$

. . . . $=\;-\ln8128 \;=\;-9.00307017$

• May 14th 2011, 10:59 AM
PapaSmurf
Quote:

Originally Posted by Soroban
Hello, PapaSmurf!

We have: . $S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128})$

. . $S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right)$

. . . . $=\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right)$

. . . . $=\;-\ln8128 \;=\;-9.00307017$

WOW, THANK YOU SO MUCH!! Dude I had no idea that you could do that with 'ln's, I completely forget about it after finals in January

Thanks again! How do I give rep to you?
• May 14th 2011, 11:08 AM
topsquark
Quote:

Originally Posted by PapaSmurf
How do I give rep to you?

You have two choices. First, "thanking" a member does give reputation. If you insist on rep, though, you cannot give rep after thanking a member. So what you will have to do is remove your thanks, then click on the six pointed star at the bottom left of the post.

-Dan