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Evaluate $\sum\limits_{n=1}^\infty\frac{n^2}{(n+2)^3}$

How can I evaluate this sum?
$$\sum_{n=1}^\infty\frac{n^2}{(n+2)^3}$$
Is there any trick? Is there any series that I can use?

A:

$$\sum_{n=1}^\infty\frac{n^2}{(n+2)^3}=\sum_{n=1}^\infty\frac{n^2}{n^3+6n^2+10n+4}=\sum_{n=1}^\infty\frac{1}{n^3+6n^2+10n+4}-\sum_{n=1}^\infty\frac{1}{n^3+7n^2+10n+4}$$
Since $7n^2+10n+4=n^2+5n+4$ and $n^3+7n^2+10n+4=n^3+11n^2+10n+4=(n+2)^3$, the sum is
$$\sum_{n=1}^\infty\frac{1}{n^3+6n^2+10n+4}=\sum_{n=1}^\infty\frac{1}{n^3+5n^2+5n+1}$$
We have that the sum
$$\sum_{n=1}^\infty\frac{1}{n^3+5n^2+5n+1}=\sum_{n=1}^\infty\frac{1}{(n+1)^3-4n^2}=\frac{1}{3}\sum_{n=1}^\infty\frac{1}{(n+1)^2}-\frac{4}{9}\sum_{n=1}^\infty\frac{n^2}{(n+1)^3}$$
Now, the first sum is $3/(1+4)=1$ and the second sum is 384a16bd22

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■ Registering an autoload keymacro for any given playlist output scheme. This can be used to synchronize any of the related functions across different instances of a playlist.
■ Disabling of the keymacro on shutdown.
■ Allows custom formatting of the playlist output. For instance; Change the title of the playlist title, as well as the colour of the playlist output.
■ Modifying the appearance of the function that creates the playlist output.
■ Modifying the way the playlist output is displayed.
■ Modifying the way that the linked files are displayed.
System Requirements:
■ foobar2000 must be version 2.2.0 or higher. (foobar2000 2.2.0.0 is required for this plugin to work).
■ The latest and greatest is available on their website.
■ foo playlist output is a registered trademark and proprietary software copyright, Foobar2000 Ltd.
How to use this plugin:
Load the plugin into your foobar2000 instance, you will see a new option appear in the File menu. Once installed, load a playlist into foobar2000 and you will see the option for the foo playlist output output appear.
Then, you will be able to change the properties of this plugin in the same menu as the foo playlist output option. This is shown below:
■ Title — This is displayed above the title of the playlist output in the main foobar2000 window
■ Colour — This determines the colour of the title of the playlist output
■ Descriptions — This is a link to a file containing the description of the playlist output in foobar2000. This can be updated at any time by editing the foo playlist output description file in the Plugin installer directory. For instance, editing the «foo-playlist.m3u» file in the foobar2000/plugins directory will update the description.
■ Format — Change the way the playlist output is displayed by using this function. A full list of supported functions can be found in the manual in the «functions» section.
How to activate this plugin:
To enable this plugin, load it into foobar2000 and enable the plugin. Then, you will see the new option available to you in the File menu.
How to uninstall this plugin:
Load the plugin into foobar

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