# [SOLVED]Simplifying a summation

#### dwsmith

##### Well-known member
u is a complex.

Can this sum be simplified? If so, how?

$\displaystyle \sum_{n}^{\infty}\frac{2u+5}{2}\left(-\frac{u}{2}\right)^n$

Thanks.

#### Also sprach Zarathustra

##### Member
u is a complex.

Can this sum be simplified? If so, how?

$\displaystyle \sum_{n}^{\infty}\frac{2u+5}{2}\left(-\frac{u}{2}\right)^n$

Thanks.

Yes.

$\sum_{n}^{\infty}\frac{2u+5}{2}(-\frac{u}{2})^n=\frac{2u+5}{2}\sum_{n}^{\infty}(-1)^n(\frac{u}{2})^n$

Can you proceed? (with $|u|<2$)

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#### dwsmith

##### Well-known member
Yes.

$\sum_{n}^{\infty}\frac{2u+5}{2}(-\frac{u}{2})^n=\frac{2u+5}{2}\sum_{n}^{\infty}(-1)^n(\frac{u}{2})^n$

Can you proceed? (with $|u|<2$)
Proceed? I am not trying to solve just simplify it. I thought it could have been expressed a lot differently since I could have done that.

Thanks.

#### Also sprach Zarathustra

##### Member
Proceed? I am not trying to solve just simplify it. I thought it could have been expressed a lot differently since I could have done that.

Thanks.

$\sum_{n}^{\infty}\frac{2u+5}{2}(-\frac{u}{2})^n=\frac{2u+5}{2}\sum_{n}^{\infty}(-1)^n(\frac{u}{2})^n=\frac{2u+5}{2}\frac{2}{u+2}=$ $\frac{2u+5}{u+2}$

(sum of geometric series)

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#### dwsmith

##### Well-known member
$\sum_{n}^{\infty}\frac{2u+5}{2}(-\frac{u}{2})^n=\frac{2u+5}{2}\sum_{n}^{\infty}(-1)^n(\frac{u}{2})^n=\frac{2u+5}{2}\frac{2}{u+2}=\frac{2u+5}{u+2}$

(sum of geometric series)
I am not needing to solve these just simplify in the summation notation.

Can this one be simplified further?

$\sum_{n = 0}^{\infty}\left[(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)\right]$

#### Also sprach Zarathustra

##### Member
I am not needing to solve these just simplify in the summation notation.

Can this one be simplified further?

$\sum_{n = 0}^{\infty}\left[(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)\right]$

$\sum_{n = 0}^{\infty}(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)=\sum_{n = 0}^{\infty}(-1)^n z^n(1-\frac{n+2}{2^n})=$

$=\sum_{n = 0}^{\infty}(-1)^n z^n-\sum_{n = 0}^{\infty}(-1)^n\frac{n+2}{2^n} z^n=$

$=\sum_{n = 0}^{\infty}(-1)^n z^n-\sum_{n = 0}^{\infty}(-1)^n n(\frac{n}{2})^n z^n-2 \sum_{n = 0}^{\infty}(-1)^n(\frac{z}{2})^n$

And each one of the sums can be evaluated.

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#### dwsmith

##### Well-known member
$\sum_{n = 0}^{\infty}(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)=\sum_{n = 0}^{\infty}(-1)^n z^n(1-\frac{n+2}{2^n})=$

$=\sum_{n = 0}^{\infty}(-1)^n z^n-\sum_{n = 0}^{\infty}(-1)^n\frac{n+2}{2^n} z^n=$

$=\sum_{n = 0}^{\infty}(-1)^n z^n-\sum_{n = 0}^{\infty}(-1)^n n(\frac{n}{2})^n z^n-2 \sum_{n = 0}^{\infty}(-1)^n(\frac{z}{2})^n$

And each one of the sums can be evaluated.

 $\sum_{n = 0}^{\infty}(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)=\sum_{n = 0}^{\infty}(-1)^n z^n(1-\frac{n+2}{2^n})$
How did you get this $\frac{n+2}{2^n}$? I can't get that.

#### Also sprach Zarathustra

##### Member
 $\sum_{n = 0}^{\infty}(-1)^n z^n \left(1 - \frac{1}{2^{n + 1}} - n\frac{1}{2^n}\right)=\sum_{n = 0}^{\infty}(-1)^n z^n(1-\frac{n+2}{2^n})$
How did you get this $\frac{n+2}{2^n}$? I can't get that.
Sorry!

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

#### soroban

##### Well-known member
Hello, dwsmith!

No one is trying "solve" anything . . . They are all trying to simplify.
Stop complaining!

$\displaystyle \text{Simplify: }\:\sum_{n=0}^{\infty}\frac{2u+5}{2}\left(-\frac{u}{2}\right)^n$

Did you even read what A.S.Z. wrote?

He said: .$\displaystyle \sum^{\infty}_{n=0}\frac{2u+5}{2}\left(-\frac{u}{2}\right)^n \;=\;(2u+5)\underbrace{\sum^{\infty}_{n=0}\left(-\frac{u}{2}\right)^n}_{\text{geometric series}}$

We have a geometric series with: first term $a = 1$ and common ratio $r = \text{-}\frac{u}{2}$

If $u < 2$, the series converges to a finite limit.

Can you find it?
Can you finish the problem?

#### dwsmith

##### Well-known member
Hello, dwsmith!

No one is trying "solve" anything . . . They are all trying to simplify.
Stop complaining!

Did you even read what A.S.Z. wrote?

You shouldn't jump to conclusions though. You don't know what my intentions are. What is useful to my focus is only the summation not the fact it is geometric.

And CB to thank someone for something that is wrong in the first place is trite because none of you knew what I was doing with the summation. I find both of these actions rude.

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#### Prove It

##### Well-known member
MHB Math Helper
You shouldn't jump to conclusions though. You don't know what my intentions are. What is useful to my focus is only the summation not the fact it is geometric.

And CB to thank someone for something that is wrong in the first place is trite because none of you knew what I was doing with the summation. I find both of these actions rude.
You'll have to forgive us, our psychic powers have been a little off since MHF died. We'll have to work on our mind-reading in future.

#### dwsmith

##### Well-known member
You'll have to forgive us, our psychic powers have been a little off since MHF died. We'll have to work on our mind-reading in future.
It isn't about being psychic it is about not assuming. Never assume you understand what someone else is doing because you may be wrong. Heuristics are great but not the end all be all.