Expert: Paul Klarreich - 11/22/2005

Question
Hi, my name is leslie and I want to know how i can seperate (2x+2)! I know that if i have (x+1)! i can seperate it to be (x+1)(x)! so how do i seperate (2x+2)!


Also, i want to know how to find the sum of these series: the summation from n=0 to infinity of [ (9/8)^n +(3/4)^n ] & the summation from n=1 to infinity of [ (-1)^(n-1) (7/5)^n ]


I have tried using the geometric series formula a/(1-r) to get the sum of both of these series but for some reason im not getting the right answer. Please help me if you can. You don't know how much I will appreciate it! Thank you for your time.

                        Sincerely,

                         Leslie  

Answer
Of course, the word is SEPARATE.  But...

You can rewrite (2x+2)! in much the same way; whether it will do you a lot of good, I don't know.

(anything)! means to write as FACTORS all the integers from (anything) down to 1.  The next integer down from (2x+2) is (2x+1), then comes (2x), (2x-1), etc.  So any of these might be possible:

(2x+2)! = (2x+2)(2x+1)!
(2x+2)! = (2x+2)(2x+1)(2x)!
(2x+2)! = (2x+2)(2x+1)(2x)(2x-1)!
etc.



Also, i want to know how to find the sum of these series: the summation from n=0 to infinity of [ (9/8)^n +(3/4)^n ]

This first one is the sum of two series:

 Sum    (9/8)^n
n=0..inf

 Sum    (3/4)^n
n=0..inf

Bad news, kemo sabe.  The first one diverges, so your sum does not exist.

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& the summation from n=1 to infinity of [ (-1)^(n-1) (7/5)^n ]

I have tried using the geometric series formula a/(1-r) to get the sum of both of these series but for some reason I'm not getting the right answer.
--------------------------
The reason is clear:  You forgot something.  You think the formula for the sum of ar^n is a/(1-r).  Wrong!  The formula is a/(1-r) provided |r| < 1.

That little bit makes all the difference.  Whenever |r| > 1, as in the first example, where r = 9/8 > 1, or in the second, where r = 7/5 > 1, the series will diverge, and the sum does not exist.

Did you miscopy the example?  Were these supposed to be 8/9 and 5/7?