14. Given that $\frac{\pi^{4}}{18}=\sum_{n=1}^{\infty} \frac{5}{n^{4}}$ what is $\sum_{n=2}^{\infty} \frac{5}{n^{4}}$ equal to?
$\frac{\pi^{4}-85}{16}$
$\frac{\pi^{4}-90}{18}$
$\frac{\pi^{4}-5}{18}$
$\frac{\pi^{4}-72}{18}$
$\frac{\pi^{4}-85}{18}$

Answer

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Answer

Final Answer: $\boxed{\frac{\pi^{4}-5}{18}}$

Steps

Step 1 ：Given that $\frac{\pi^{4}}{18}=\sum_{n=1}^{\infty} \frac{5}{n^{4}}$, we are asked to find $\sum_{n=2}^{\infty} \frac{5}{n^{4}}$.

Step 2 ：We can find this by subtracting the first term of the series (when n=1) from the total sum.

Step 3 ：The first term of the series is $\frac{5}{1^{4}} = 5$.

Step 4 ：So, $\sum_{n=2}^{\infty} \frac{5}{n^{4}} = \frac{\pi^{4}}{18} - 5$.

Step 5 ：Final Answer: $\boxed{\frac{\pi^{4}-5}{18}}$