# 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!

Mentor
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}##.
Frobenius21 said:
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.

Mentor
Thanks a lot, I am asked to find the sum of the series not to show if it converges.
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.

Homework Helper
2022 Award
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

Homework Helper
Gold Member
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

Mentor
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.

#### Attachments

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