Thread: Sum to n of series

1. Sum to n of series

If $\displaystyle S_n=\displaystyle \sum^n_{r=0} a^r(1+a+a^2+...+a^r)$, $\displaystyle (|a|\neq 1)$ by considering $\displaystyle (1-a)S_n$, show that
$\displaystyle S_n=\frac{1-a^{2n+2}}{(1-a^2)(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$
State the set of values of a for which $\displaystyle S_n$approaches a limit as $\displaystyle n \rightarrow\infty$ and find the sum to infinity of the series for these values.

Attempt:
$\displaystyle \displaystyle \sum^n_{r=0} a^r(1+a+a^2+...+a^r)$
$\displaystyle =\displaystyle \sum^n_{r=0} a^r\left(\displaystyle \sum^n_{r=0}a^r\right)$
$\displaystyle =\displaystyle \sum^n_{r=0}a^r \left(\frac{1-a^r}{1-a}\right)$
$\displaystyle =\displaystyle \sum^n_{r=0}\frac{a^r-a^{2r}}{1-a}$
Am I correct so far? and how to continue because I don't see a way of summing that?
Thanks!

2. This is almost correct. Here is the correct version:

$\displaystyle \displaystyle \sum^n_{r=0} a^r\left(\displaystyle \sum^r_{k=0}a^k\right)$

$\displaystyle =\displaystyle \sum^n_{r=0}a^r \left(\frac{1-a^{r+1}}{1-a}\right)$

In the question you are told to consider $\displaystyle (1-a)S_n$. Why don't you try that!

3. $\displaystyle (1-a)S_n=S_n-aS_n$
$\displaystyle =\displaystyle \sum^n_{r=0}a^r\left(\frac{(1-a^{r+1})(1-a)}{1-a}\right)$
$\displaystyle =\displaystyle \sum^n_{r=0}a^r(1-a^{r+1})$
$\displaystyle \displaystyle \sum^n_{r=0}a^r-\sum^n_{r=0}a^{2r+1}$
$\displaystyle =\frac{1-a^{n+1}}{1-a}-\frac{a-a^{2n+2}}{1-a^2}$

$\displaystyle S_n=\frac{1-a^{n+1}}{(1-a)^2}-\frac{a-a^{2n+2}}{(1-a^2)(1-a)}$
Which isn't the given one and I can't see where I've gone wrong
Thanks!

4. Originally Posted by arze
If $\displaystyle S_n=\displaystyle \sum^n_{r=0} a^r(1+a+a^2+...+a^r)$, $\displaystyle (|a|\neq 1)$ by considering $\displaystyle (1-a)S_n$, show that
$\displaystyle S_n=\frac{1-a^{2n+2}}{(1-a)^2(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$
State the set of values of a for which $\displaystyle S_n$approaches a limit as $\displaystyle n \rightarrow\infty$ and find the sum to infinity of the series for these values.

Attempt:
$\displaystyle \displaystyle \sum^n_{r=0} a^r(1+a+a^2+...+a^r)$
$\displaystyle =\displaystyle \sum^n_{r=0} a^r\left(\displaystyle \sum^n_{r=0}a^r\right)$
$\displaystyle =\displaystyle \sum^n_{r=0}a^r \left(\frac{1-a^r}{1-a}\right)$
$\displaystyle =\displaystyle \sum^n_{r=0}\frac{a^r-a^{2r}}{1-a}$
Am I correct so far? and how to continue because I don't see a way of summing that?
Thanks!
Dear arze,

When n=1,

$\displaystyle S_1=\displaystyle\sum_{r=0}^{1}{a^r(1+a+a^2+.....+ a^r)}=a^0+a(1+a)$

But if you consider the expression, $\displaystyle \frac{1-a^{2n+2}}{(1-a)^2(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$......

When n=1,

$\displaystyle \frac{1-a^{2n+2}}{(1-a)^2(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$

$\displaystyle =\frac{1-a^{4}}{(1-a)^3}-\frac{a^{2}(1+a^{2})}{(1-a)^2}$

$\displaystyle =\frac{1+a^2}{1-a^2}\left(\frac{1-a^2}{1-a}-a^2\right)$

$\displaystyle =\frac{1+a^2}{1-a^2}\left(1+a-a^2\right)$

Therefore, $\displaystyle S_1\neq\frac{1-a^{2n+2}}{(1-a)^2(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$

5. Yea, that's not the correct identity. The correct identity is

[tex]\displaystyle \displaystyle S_n = \frac{(1-a^{n+1})(1-a^{n+2})}{(1-a)^2(1+a)}[/Math]

6. Originally Posted by Vlasev
Yea, that's not the correct identity. The correct identity is

[tex]\displaystyle \displaystyle S_n = \frac{(1-a^{n+1})(1-a^{n+2})}{(1-a)^2(1+a)}[/Math]
But that doesn't give me the result I'm supposed to get either
$\displaystyle S_n=\frac{1-a^{2n+2}}{(1-a^2)(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$

7. Originally Posted by arze
But that doesn't give me the result I'm supposed to get either
$\displaystyle S_n=\frac{1-a^{2n+2}}{(1-a^2)(1-a)}-\frac{a^{n+1}(1+a^{n+1})}{(1-a)^2}$
Because what you are asked to get is (in this form) false! Unless you typed something incorrectly in your question.

8. hmmm, i checked the book again and i did not type anything wrong. Well, thank you very much!

9. $\displaystyle \displaystyle\ S_n=\sum_{r=0}^na^r\left(1+a+a^2+....+a^r\right)$

$\displaystyle \displaystyle\ aS_n=\sum_{r=0}^na^{r+1}\left(1+a+a^2+....+a^r\rig ht)$

$\displaystyle \displaystyle\ (1-a)S_n=\sum_{r=0}^n\left(a^r+a^{r+1}+a^{r+2}+....+a ^{2r}\right)-\sum_{r=0}^n\left(a^{r+1}+a^{r+2}+....+a^{2r+1}\ri ght)$

$\displaystyle \displaystyle\ =\sum_{r=0}^na^r-\sum_{r=0}^na^{2r+1}$

The first sum has $\displaystyle n+1$ terms, the first term is 1 and the common ratio is $\displaystyle a$.
The second sum has $\displaystyle (2n+2)/2 =n+1$ terms, the first term is $\displaystyle a$ and the common ratio is $\displaystyle a^2$.

$\displaystyle \displaystyle\ (1-a)S_n=\frac{1-a^{n+1}}{1-a}-a\left(\frac{1-a^{2(n+1)}}{1-a^2}\right)$

$\displaystyle \displaystyle\ S_n=\frac{1-a^{n+1}}{(1-a)^2}-a\left(\frac{1-a^{2n+2}}{(1-a)^2(1+a)}\right)$

$\displaystyle \displaystyle\huge\ =\frac{(1+a)\left(1-a^{n+1}\right)-a\left(1-a^{2n+2}\right)}{(1-a)^2(1+a)}$

$\displaystyle \displaystyle\huge\ =\frac{(1+a)\left(1-a^{n+1}\right)-a\left(1-a^{n+1}\right)\left(1+a^{n+1}\right)}{(1-a)^2(1+a)}$

$\displaystyle \displaystyle\huge\ =\frac{\left(1-a^{n+1}\right)\left[1+a-a\left(1+a^{n+1}\right)\right]}{(1-a)^2(1+a)}$

$\displaystyle \displaystyle\huge\ =\frac{\left(1-a^{n+2}\right)\left(1-a^{n+1}\right)}{(1-a)^2(1+a)}$

so I'd have to agree with Vlasev.