# Rearranging Series

by andrey21
Tags: rearranging, series
 Mentor P: 16,593 Yes, this is correct. So what are your values for A and B? What is $$\frac{1}{(n+1)(n+4)}=...$$?
 P: 466 A = 1/3 b= -1/3
 Mentor P: 16,593 Yes, so we have the series $$\sum_{n=1}^{+\infty}{\frac{1}{3(n+1)}-\frac{1}{3(4+n)} }$$ Now you have to use a little trick. Try to write out this series for the first 10 terms (without adding any of the fractions). Do you see something cool?
 P: 466 Alot of the terms cancel out and im left with: 1/3+ 1/6+1/9 -1/36-1/39-1/42 for first 10 terms:
 Mentor P: 16,593 Yes, that's exactly what we wanted! Can you do this in general now? If you calculate the first k terms, there will be a lot of terms cancelling out. So which of the terms remain? If you don't see this immediately, try taking k some other values. Try taking k=20 and k=30. Then you will be ready to handle the general situation...
 P: 466 Well in this case it is the first 3 and last 3 terms which remain correct?
 Mentor P: 16,593 Indeed. So you've shown that $$\sum_{n=1}^{k}{\left(\frac{1}{3(n+1)}-\frac{1}{3(n+4)}\right)}=\frac{1}{3}+\frac{1}{6}+\frac{1}{9}-\frac{1}{3(k+2)}-\frac{1}{3(k+3)}-\frac{1}{3(k+4)}$$ Now you can easily take the limit $$k\rightarrow +\infty$$
 P: 466 so as k tends to infinty the limit is 11/18.
 Mentor P: 16,593 Correct!! It seems like you've got it!
 P: 466 Fantastic thanks micromass. I do have one final question very similar. given the sequence: 1+1/9+1/25+1/49.... I know this can be written as: sum 1/(2n+1)^2 correct?? What does this converge to??
 Mentor P: 16,593 Do you know what the sum of $$\sum_{n=1}^{+\infty}{\frac{1}{n^2}}$$ is?? If you don't know the above sum, then your question is very difficult...
 P: 466 Yes the sum of 1/n^2 is pi^2/6 correct??
 Mentor P: 16,593 Ah, yes. this is good. Now, you've got that $$\sum_{n=1}^{+\infty}{\frac{1}{n^2}}=\sum_{n=1}^{+\infty}{\frac{1}{(2n)^ 2}}+\sum_{n=1}^{+\infty}{\frac{1}{(2n+1)^2}}$$ You know two of the above series...
 P: 466 ah yes i see simply solve to get: 1/4n^2 which equals pi^2/24 correct?? Then subtract to leave me value for 1/(2n+1)^2
 Mentor P: 16,593 Correct!
 P: 466 Great thanks micromass :)
 P: 466 I do have another series question micromass: I have bin given the series: 5/4 + 1 + 4/5 + 16/25+.... It says describe what type of series this is? Shall i try find general formula again?
 Mentor P: 16,593 The series is special type of serie. What special kinds of sequences/series have you seen? Edited because I made a mistake somewhere (:

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