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anemone
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Evaluate \(\displaystyle \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}\).
anemone said:Evaluate \(\displaystyle \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}\).
MarkFL said:My solution:
Let's write the sum as:
\(\displaystyle S_n=\sum_{k=2}^{n}\left(\frac{3k^2+1}{\left(k^3-k\right)^3}\right)\)
Performing a partial fraction decomposition on the summand, and using the properties of sums, we obtain:
\(\displaystyle S_n=\frac{1}{2}\sum_{k=2}^{n}\left(\frac{1}{(k+1)^3}-\frac{2}{k^3}+\frac{1}{(k-1)^3}\right)+\frac{3}{2}\sum_{k=2}^{n}\left(\frac{1}{(k+1)^2}-\frac{1}{(k-1)^2}\right)+3\sum_{k=2}^{n}\left(\frac{1}{k+1}-\frac{2}{k}+\frac{1}{k-1}\right)\)
Let's now look at the 3 sums on the right in turn:
\(\displaystyle \sum_{k=2}^{n}\left(\frac{1}{(k+1)^3}-\frac{2}{k^3}+\frac{1}{(k-1)^3}\right)=1+\frac{1}{8}-\frac{1}{4}+\sum_{k=4}^{n-2}\left(\frac{1}{k^3}-\frac{2}{k^3}+\frac{1}{k^3}\right)+\frac{1}{(n+1)^3}+\frac{1}{n^3}-\frac{2}{n^3}=\frac{7}{8}+\frac{1}{(n+1)^3}-\frac{1}{n^3}\)
\(\displaystyle \sum_{k=2}^{n}\left(\frac{1}{(k+1)^2}-\frac{1}{(k-1)^2}\right)=-1-\frac{1}{4}+\sum_{k=4}^{n-2}\left(\frac{1}{k^2}-\frac{1}{k^2}\right)+\frac{1}{(n+1)^2}+\frac{1}{n^2}=-\frac{5}{4}+\frac{1}{(n+1)^2}+\frac{1}{n^2}\)
\(\displaystyle \sum_{k=2}^{n}\left(\frac{1}{k+1}-\frac{2}{k}+\frac{1}{k-1}\right)=1+\frac{1}{2}-\frac{2}{2}+\sum_{k=4}^{n-2}\left(\frac{1}{k}-\frac{2}{k}+\frac{1}{k}\right)+\frac{1}{n+1}+\frac{1}{n}-\frac{2}{n}=\frac{1}{2}+\frac{1}{n+1}-\frac{1}{n}\)
And so, we may now state the partial sum as:
\(\displaystyle S_n=\frac{1}{2}\left(\frac{7}{8}+\frac{1}{(n+1)^3}-\frac{1}{n^3}\right)+\frac{3}{2}\left(-\frac{5}{4}+\frac{1}{(n+1)^2}+\frac{1}{n^2}\right)+3\left(\frac{1}{2}+\frac{1}{n+1}-\frac{1}{n}\right)=\frac{(n(n+1))^3-8}{16(n(n+1))^3}\)
And so the infinite sum is:
\(\displaystyle S_{\infty}=\lim_{n\to\infty}\left(S_n\right)=\frac{1}{16}\)
To evaluate a complex sum, you need to break it down into smaller parts and simplify them individually. You can also use mathematical rules and operations such as addition, subtraction, multiplication, and division to solve the complex sum.
Some common strategies for evaluating complex sums include breaking them down into simpler parts, using mathematical rules and operations, and using algebraic manipulation to simplify the sum.
One of the main challenges in evaluating complex sums is identifying the correct mathematical operations and rules to use. Another challenge is keeping track of all the steps and calculations involved in simplifying the sum.
You can check the correctness of your solution by plugging it back into the original complex sum and seeing if it results in the same value. Another way is to use a calculator or software program to verify the solution.
No, mathematical rules and operations are necessary for evaluating complex sums. However, there are other methods such as using graphing calculators or software programs that can assist in solving complex sums without manually applying mathematical rules and operations.