A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 2 years ago
If I need to find a closed solution for the summation of x*a^x from x = 0 to n, what do I do? For starters, I've decided to do an integral from 0 to b of the same function, but now I have 2 variables! ah! Help please :D
 2 years ago
If I need to find a closed solution for the summation of x*a^x from x = 0 to n, what do I do? For starters, I've decided to do an integral from 0 to b of the same function, but now I have 2 variables! ah! Help please :D

This Question is Closed

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0The only hint I was given was to take the integral of the function I was trying to take the summation of. But, now I have\[\int\limits_{0}^{b}x*a^x da\]

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0I can't seem to do a usub that will work for me.

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0I want to assume that x is a constant and pull it outside, but I don't know if I can do that.

extremity
 2 years ago
Best ResponseYou've already chosen the best response.0you can, since youre differentiating with respect to a.

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0Because S = (0)a^(0) + (1)a^(1)...+(n)a^(n); you see why I think it's a constant?

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0Yes! But what happens to the other x?

extremity
 2 years ago
Best ResponseYou've already chosen the best response.0wel, the other x is a constant too right?

extremity
 2 years ago
Best ResponseYou've already chosen the best response.0whats the integral of a^x , if x is constant?

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1I have a wonderful solution to this, I believe.

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0hm, you are right extremity

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{0}^{b}a^x da\]?

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \sum_{x=0}^n xa^{\lambda x} = \sum_{x=0}^n \frac{d}{d\lambda} \frac{a^{\lambda x}}{\ln(a)} = \frac{1}{\ln(a)} \frac{d}{d\lambda} \sum_{x=0}^n a^{\lambda x}\] \[ = \frac{1}{\ln(a)} \frac{d}{d\lambda} \frac{1 a^{\lambda (n+1)}}{1 a^\lambda}\]

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0where'd you get lambda?

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1just made it up. Set it to 1 when you're all done.

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0it's to differentiate x from the exponent?

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1yep. I may have made a typo, but I'm getting \[ \frac{a(n+1) a^n  n}{(1a)^2} \] as a final result. Assuming of course that a does not equal 1.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1Oops, I did make a typo

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1should be \[ \frac{na^{n+2}  na^{n+1}  a^{n+1} + a}{(1a)^2} \]

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1And wolfram alpha confirms. Awesome.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1By the way, if a = 1, obviously the sum is just \[ \sum_{x=0}^n x = \frac{n(n+1)}{2}\]

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0Thanks! I am going back over this to make sure I understand. Thanks for all your help!

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1No problem. Be careful with your derivatives. And @TuringTest / @Gravion should recognize this technique.... ;)

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0I have a question: what if I need to use the hint (i.e., that's what they want us to do). I kinda see what you are doing, but I can move on from the integral from 0 to infinity of x*a^x

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0the integral doesn't converge, so...

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1That is the hint. I integrated, then differentiated.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1And it should converge if a is sufficiently small.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1If a < 1, that reduces to \[ \frac{a}{(1a)^2} \]

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0I'm sorry, I'm thrown off by the ln(a). I can't get it in my own work...

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0I'm getting a^(x+1)/ (x+1) though, which I can use on the limits of intergration...but then I get x[infinity  a] which doesn't make sense!

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1Did you follow my reasoning up until the part where you take the derivative?

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0um, actually no. I don't understand d/dlambda

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0you said you made lamda upto differentiate from the other x (I guess) but then how do you also do a derivative for it?

ilovemath7
 2 years ago
Best ResponseYou've already chosen the best response.0wait, the "integrate then differentiate" thing is what I am supposed to do (that's the hint they gave" but I dunno why I can't understand this!

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1I invented a parameter to put in the problem, differentiated and integrated with respect to said parameter, and at the end of the day, after everything was done, I set that parameter to 1.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1It's not a trivial series to sum, what class are you doing this for?
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.