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help finding infinite series

 
 
how do I find the sum of the follwoing infinite series.  Please show work/steps:
 
  1.  1/2 + 1 + 2 + 4 ........
 
  2.  9-3 + 1 - 1/3 + 1/9 ......
 
  3. How do I write the first three terms of the expansion of (a-b)^6
 
 4.  how do I find the last three terms of (a - b)^7
 
 
thanks
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1 Answer

Dear L,
this is a lot of problems to be having trouble with, all at once....
For (1), what's happening to the partial sums as you add more terms from the series? Is it ever going to settle down?
For (2), write your series as x = the series. Then, multiply that whole equation by some value (lets's call it "a"). You then can write a new equation:
ax = a(9) -a(3) + (a)1 - .....     such that you get a bunch of terms on the right side that look like some of your original x series terms, but they have the opposite sign. Try it for some different values of a, both positive and negative! Hint: what is the quotient between two successive terms in your series?
When you do get an equation for "ax" that has terms on the right side that look like some of your original x series terms, but have the opposite sign, you can then ADD your two equations to each other. How do you do that? Just add the things on the left, write it down, write the = sign, then add the things on the right, write that down. Adding the right should now be easy -- for just about every term in the first equation, there'll be a term in the second equation canceling it out when added (such as, a -3 will cancel a +3, and so on). Only essentially the first term of one of the equations will be left, on the right (and the end term of the other equation, but that's trending towards zero so we can ignore it). Then do the math, to solve for "x". You've found the value of the infinite "geometric" series!
Questions (3) and (4) are asking you for coefficients and appropriate term powers in the binomial expansion. If you're clueless on what that is, look it up, and start figuring the successive (a-b) powers longhand, until you get a feel for what's going on. You should also draw (if you haven't already, in class) a binomial pyramid of coefficients, and compare it to what you got figuring the powers of (a-b) longhand.
Hope this gets you going,
-- S.

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