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I can't seem to do a u-sub that will work for me.

I want to assume that x is a constant and pull it outside, but I don't know if I can do that.

you can, since youre differentiating with respect to a.

Because S = (0)a^(0) + (1)a^(1)...+(n)a^(n); you see why I think it's a constant?

Yes! But what happens to the other x?

wel, the other x is a constant too right?

whats the integral of a^x , if x is constant?

I have a wonderful solution to this, I believe.

Bring it :D

hm, you are right extremity

\[\int\limits_{0}^{b}a^x da\]?

\[a^{x+1} /(x+1)\]

woah.

where'd you get lambda?

just made it up. Set it to 1 when you're all done.

it's to differentiate x from the exponent?

Oops, I did make a typo

should be
\[ \frac{na^{n+2} - na^{n+1} - a^{n+1} + a}{(1-a)^2} \]

And wolfram alpha confirms. Awesome.

By the way, if a = 1, obviously the sum is just
\[ \sum_{x=0}^n x = \frac{n(n+1)}{2}\]

lol

Thanks! I am going back over this to make sure I understand. Thanks for all your help!

:D

Are you still on?

the integral doesn't converge, so...

That is the hint. I integrated, then differentiated.

And it should converge if a is sufficiently small.

hm...

If |a| < 1, that reduces to
\[ \frac{a}{(1-a)^2} \]

I'm sorry, I'm thrown off by the ln(a). I can't get it in my own work...

Did you follow my reasoning up until the part where you take the derivative?

um, actually no. I don't understand d/dlambda

It's not a trivial series to sum, what class are you doing this for?