How do i convert $\frac{1}{2+x}$ to a summation?

Question

I am given the summation for $\frac{1}{1-x}$. I get that I need to sub in $-x$ for $x$. I don't get how I am supposed to know where I put the $2$.

I am not sure if there is a systematic procedure or if I am just lacking analytical skill.

Thanks in advance!

Factor out the 2 from the denominator, to get $\frac{1}{2(1-(-(x/2)})$ and treat it as the sum of a geometric progression. — user99680, Nov 06 '13 at 00:17
Does that mean I have 1/2 as a coefficient of sigma, and i have negative half x where x should be? — Tad, Nov 06 '13 at 00:22
:please see my answer below and let me know if you have a question. — user99680, Nov 06 '13 at 00:23
Let x = -1-u. Then 1/(2+x) = 1/(2-1-u) = 1/(1-u). Don't worry about your analytic skill -- it will grow. — Betty Mock, Nov 06 '13 at 00:24
@yanbo if you're going to edit the question, make sure you read it. — Dan Rust, Nov 06 '13 at 00:48

score 2 · Accepted Answer · answered Nov 06 '13 at 00:42

2

Consider $${1\over (2+x)}={1\over 2(1-({-x\over 2}))}.$$ This is now in the form of the geometric series. So $${1\over 2}\cdot {1\over (1-({-x\over 2})}={1\over 2}\cdot \sum_{n=0}^\infty ({-x\over 2})^n=\sum_{n=0}^\infty{(-1)^n\over 2^{n+1}}x^n.$$

answered Nov 06 '13 at 00:42

1233dfv

5,625

Is there a reason why x^n is written out of the fraction? Also, why did you separate (-1)^n ? – Tad Nov 06 '13 at 00:59
No reason really, just something I always do. I keep constants on the left. I did separate it. We had $(-x)^n=(-1)^nx^n$. – 1233dfv Nov 06 '13 at 01:02

user99680 · Answer 2 · 2013-11-06T00:32:44.940

0

Using the fact that for $|r|$<1 , we have:

$$ \frac{1}{1-r}=1-r+r^2+......$$

Rewrite $\frac{1}{2+x}$ as $\frac{1}{2(1+x/2)}=\frac{1}{2}\frac{1}{1-(-x/2)}$, and

apply the above identity.

edited Nov 06 '13 at 00:32

answered Nov 06 '13 at 00:21

user99680

6,708

Several of the $-$ signs should be $+$ (except for the left-hand-side of the above identity with $r$s). – Patterns_43 Nov 06 '13 at 00:24
You're right, let me rewrite. – user99680 Nov 06 '13 at 00:31

How do i convert $\frac{1}{2+x}$ to a summation?

2 Answers2