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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
 one year ago
 one year 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
 one year ago
 one year ago

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ilovemath7Best ResponseYou've already chosen the best response.0
The 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\[\int\limits_{0}^{b}x*a^x da\]
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
I can't seem to do a usub that will work for me.
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
I want to assume that x is a constant and pull it outside, but I don't know if I can do that.
 one year ago

extremityBest ResponseYou've already chosen the best response.0
you can, since youre differentiating with respect to a.
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
Because S = (0)a^(0) + (1)a^(1)...+(n)a^(n); you see why I think it's a constant?
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
Yes! But what happens to the other x?
 one year ago

extremityBest ResponseYou've already chosen the best response.0
wel, the other x is a constant too right?
 one year ago

extremityBest ResponseYou've already chosen the best response.0
whats the integral of a^x , if x is constant?
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
I have a wonderful solution to this, I believe.
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
hm, you are right extremity
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
\[\int\limits_{0}^{b}a^x da\]?
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
\[ \sum_{x=0}^n xa^{\lambda x} = \sum_{x=0}^n \frac{d}{d\lambda} \frac{a^{\lambda x}}{\ln(a)} = \frac{1}{\ln(a)} \frac{d}{d\lambda} \sum_{x=0}^n a^{\lambda x}\] \[ = \frac{1}{\ln(a)} \frac{d}{d\lambda} \frac{1 a^{\lambda (n+1)}}{1 a^\lambda}\]
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
where'd you get lambda?
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
just made it up. Set it to 1 when you're all done.
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
it's to differentiate x from the exponent?
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
yep. 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.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
Oops, I did make a typo
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
should be \[ \frac{na^{n+2}  na^{n+1}  a^{n+1} + a}{(1a)^2} \]
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
And wolfram alpha confirms. Awesome.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
By the way, if a = 1, obviously the sum is just \[ \sum_{x=0}^n x = \frac{n(n+1)}{2}\]
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
Thanks! I am going back over this to make sure I understand. Thanks for all your help!
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
No problem. Be careful with your derivatives. And @TuringTest / @Gravion should recognize this technique.... ;)
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
I 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
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
the integral doesn't converge, so...
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
That is the hint. I integrated, then differentiated.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
And it should converge if a is sufficiently small.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
If a < 1, that reduces to \[ \frac{a}{(1a)^2} \]
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
I'm sorry, I'm thrown off by the ln(a). I can't get it in my own work...
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
I'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!
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
Did you follow my reasoning up until the part where you take the derivative?
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
um, actually no. I don't understand d/dlambda
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
you said you made lamda upto differentiate from the other x (I guess) but then how do you also do a derivative for it?
 one year ago

ilovemath7Best ResponseYou've already chosen the best response.0
wait, 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!
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
I 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.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.1
It's not a trivial series to sum, what class are you doing this for?
 one year ago
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