How do I find the sum of a series in terms of n?

[vin-math]

Can anyone teach me how to find the sum of the series in terms of n in the following:

1+2x3^2+3x3^4+...+(n+1)3^2n

Thx!

 

[0rthodontist]

This can be done with generating functions if you know them.
1. Find the generating function for 1, 3^2, 3^4, ...
2. From this, find the generating function for 1, 2*3^2, ...
3. Find the generating function for (n+2)*3^(2(n+1)), (n+3)*3^(2(n+2)), ...
4. Subtract the latter from the former and evaluate at 1.

 

[vin-math]

This can be done with generating functions if you know them.
1. Find the generating function for 1, 3^2, 3^4, ...
2. From this, find the generating function for 1, 2*3^2, ...
3. Find the generating function for (n+2)*3^(2(n+1)), (n+3)*3^(2(n+2)), ...
4. Subtract the latter from the former and evaluate at 1.

i don't know what generating function is but what i know is to use the summation sign to do this kind of question. i still can't do it...

 

[shmoe]

You might find it easier to try and sum:

$$1+2x^2+3x^4+\ldots+(n+1)x^{2n}$$

 

[shmoe]

You might find it easier to try and sum:

$$1+2x^2+3x^4+\ldots+(n+1)x^{2n}$$

It's not quite a geometric series, but can you turn it into one?

 

[0rthodontist]

Okay--what sums can you do that might be relevant?

 

[vin-math]

You might find it easier to try and sum:

$$1+2x^2+3x^4+\ldots+(n+1)x^{2n}$$

It's not quite a geometric series, but can you turn it into one?

i just can do till this step:

n+1 E(Sigma) r=1 (r*x^(r-1))

 

[vin-math]

Okay--what sums can you do that might be relevant?

I can do the summation of x, x^2 ...x^n, x(x+1),x(x+1)(x+2)...

actually this is my first time to touch this kind of math:)

 

[0rthodontist]

Well--
x + x^2 + ... + x^n
Take the derivative with respect to x, both term-by-term and in the sum you know.