Sums with twists

by sonone, Apr 11, 2020, 12:47 PM

I am going to find a general formula for sums like this: $1(a)+2(ar)+3(ar^2)+\cdots$ Where $|r|<1$ .

Let's call the sum $S$ . The troublesome part of this sum is the $1,2,3,\cdots$ . To get rid of the coefficients, let's multiply $S$ by $r$ :
$\begin{align*} S&=1(a)+2(ar)+3(ar^2)+4(ar^3)+\cdots\\ \vspace{55pt} Sr&= \phantom{1(a).....}1(ar)+2(ar^2)+3(ar^3)+\cdots \end{align*}$ Now let's eliminate some terms by computing $S-Sr$ : $\begin{align*} S&=1(a)+2(ar)+3(ar^2)+4(ar^3)+\cdots\\ Sr&= \phantom{1(a).....}1(ar)+2(ar^2)+3(ar^3)+\cdots\\ \hline S-Sr&=~~a~~+~~ar~~~+~~ar^2~~+~~ar^3~~+~~\cdots \end{align*}$ Now we can apply the infinite Geometric series formula getting $S-Sr=\dfrac{a}{1-r}.$ Solving for $S$ , we get $\begin{align*} S-Sr&=\dfrac{a}{1-r}\\ S(1-r)&=\dfrac{a}{1-r}\\ S&=\dfrac{a}{(1-r)^2}\text{ or }\dfrac{a}{1-2r+r^2}. \end{align*}$

Now try to solve these:
$\bullet ~~$ Calculate: $\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{3}8+\cdots+\dfrac{n}{2^n}+\cdots.$ $\bullet ~~$ Calculate: $\dfrac{1}{3}+\dfrac{1}{3}+\dfrac{1}{4}+\cdots+\dfrac{n}{3\cdot 2^{n-1}}+\cdots.$ $\star ~~$ What is the expected number of rolls it takes to roll a $3$ on a fair $6$ -sided die?

This post has been edited 1 time. Last edited by sonone, Apr 11, 2020, 12:47 PM
Reason: missing period not inserted

algebra

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I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM

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Sums with twists

by sonone, Apr 11, 2020, 12:47 PM

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