Sum of a series from n=1 to infinity of n^2/(2+1/n)^n

[Frobenius21]

Homework Statement

Determine the sum of the series

Sum from n=1 to infinity of n^2/(2+1/n)^n

Relevant Equations

n^2/(2+1/n)^n

I tried to write it as n^2/2^n (1+1/2n)^n
But I am stuck there and don't know what to try next.Thanks for any help in advance!

 

[Mark44]

Homework Statement:: Determine the sum of the series

Sum from n=1 to infinity of n^2/(2+1/n)^n
Relevant Equations:: n^2/(2+1/n)^n

I tried to write it as n^2/2^n (1+1/2n)^n

You need more parentheses. As you wrote it, it would be interpreted as $\frac{n^2}{2^n}(1 + 1/2 \cdot n)^n$. What I think you meant was this:
$\frac{n^2}{2^n(1 + \frac 1 {2n})^n}$.

But I am stuck there and don't know what to try next.Thanks for any help in advance!

What tests do you know that you can use to test for convergence?

 

[Frobenius21]

You need more parentheses. As you wrote it, it would be interpreted as $\frac{n^2}{2^n}(1 + 1/2 \cdot n)^n$. What I think you meant was this:
$\frac{n^2}{2^n(1 + \frac 1 {2n})^n}$.
What tests do you know that you can use to test for convergence?

Thanks a lot, I am asked to find the sum of the series not to show if it converges.
How is possible to start with that?

 

[Mark44]

Thanks a lot, I am asked to find the sum of the series not to show if it converges.
How is possible to start with that?

If you're asked to show the sum of the series, then the series must necessarily be convergent.
Offhand, I don't have any ideas. Are there any similar examples in your textbook, or has your instructor shown you some similar examples in class?

A simple-minded way to find the sum of a series is to add a bunch of terms. If the series converges at a reasonable rate, you can get a pretty good estimate of the sum of the entire series. By adding up the first 25 terms (in Excel), I get a partial sum around 3.77.

 

[Frobenius21]

If you're asked to show the sum of the series, then the series must necessarily be convergent.
Offhand, I don't have any ideas. Are there any similar examples in your textbook, or has your instructor shown you some similar examples in class?

A simple-minded way to find the sum of a series is to add a bunch of terms. If the series converges at a reasonable rate, you can get a pretty good estimate of the sum of the entire series. By adding up the first 25 terms (in Excel), I get a partial sum around 3.77.

Yes it should converge.
There are no similar examples that I know of. I am looking in textbooks to try to find something similar.

 

[pasmith]

I'm tempted to regard this as $f(1)$ where
$$f(x) = \sum_{n=1}^\infty \frac{n^2}{(1 + \frac1{2n})^n}\left(\frac x2\right)^n$$

 

[Frobenius21]

I'm tempted to regard this as $f(1)$ where
$$f(x) = \sum_{n=1}^\infty \frac{n^2}{(1 + \frac1{2n})^n}\left(\frac x2\right)^n$$

Thanks

 

[epenguin]

Can we have confirmation of what the question is - written out please.

 

[Frobenius21]

Can we have confirmation of what the question is - written out please.

Hi, I was just told to get the sum from n=1 to infinity of n^2/(2+1/n)^n

 

[Mark44]

Hi, I was just told to get the sum from n=1 to infinity of n^2/(2+1/n)^n

Does "get the sum" mean "get the exact value of the sum" or "get an approximate value of the sum"?
I don't have any ideas about how to get the exact value, but an approximate value is 3.77, as I described in post #4. @pasmith gave a hint earlier, but I don't see how that helps you get the exact value of the sum -- it's not a series that I recognize off the top of my head.

 

[Frobenius21]

Does "get the sum" mean "get the exact value of the sum" or "get an approximate value of the sum"?
I don't have any ideas about how to get the exact value, but an approximate value is 3.77, as I described in post #4. @pasmith gave a hint earlier, but I don't see how that helps you get the exact value of the sum -- it's not a series that I recognize off the top of my head.

Thanks, I am just being ask to find a way to get the number of the sum

I also used a calculator to get the result it is 3.77381 but I am being ask to use a method to get to that result.

 

[benorin]

Not sure any of this will help, but I played with the function $f(x):=\sum_{n=1}^{\infty}n^2\left( 1+\tfrac{1}{2n}\right) ^{-n}\left( \tfrac{x}{2}\right) ^n$. Note that $f(x)$ satisfies the integral equation

$$\int_{x_1=0}^{x}\int_{x_2=0}^{x_1}f(x_2 )\tfrac{dx_2 dx_1}{x_2 x_1}=\sum_{n=1}^{\infty}\left( 1+\tfrac{1}{2n}\right) ^{-n}\left( \tfrac{x}{2}\right) ^n$$

If somebody recognizes that sum on the right which *might* be a little nicer than the one we started with, "have integral equation, will travel..." Not sure where to go from here... we could also try a DE relation, but I'm stuck, I went in circles with the integral equation (proving something I already knew to start out with, oof!).