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I know that I can just add values (5 + 5/16 + 5/81 + ...), but is there any way of finding the value without having to chuck numbers into the calculator 10 times for each k ?

Thank you. :)

Edit: The reason I'm asking this is because I am trying to compute the upper and lower bounds for the following problem which I have worked on so far.

Problem Part a.

Problem Part b, c, d

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  • $\begingroup$ I suspect not... $\endgroup$
    – Angina Seng
    Commented Oct 27, 2017 at 2:36
  • $\begingroup$ Here’s the MathJax tutorial $\endgroup$
    – gen-ℤ ready to perish
    Commented Oct 27, 2017 at 2:42
  • $\begingroup$ @JasSingh Hi, Jas! Thank you for expressing your concern! Things at MSE are a bit different than other sites, such as *Yahoo! Answers*—we are far more prestigious and have a system of up-voting, down-voting, and flagging. We use this system to enforce our expectations for all posts, which are outlined in how to ask a good question. Part of this standard is that posts should be formatted clearly and in an effective or aesthetic manner. Though some people might click on those links, no one is going to sort through digitized chicken scratch. Please use MathJax. $\endgroup$
    – gen-ℤ ready to perish
    Commented Oct 27, 2017 at 3:43

1 Answer 1

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You can have a good approximation if you notice that $$S_n=\sum_{i=1}^n \frac 1 {k^4}=H_n^{(4)}$$ where appear generalized harmonic numbers.

For large values of $n$,

$$H_n^{(4)}=\frac{\pi ^4}{90}-\frac{1}{3 n^3}+\frac{1}{2 n^4}-\frac{1}{3 n^5}+O\left(\frac{1}{n^6}\right)$$ making $$S_{10}\approx \frac{\pi ^4}{90}-\frac{43}{150000}\approx \color{red}{1.0820365}67$$ while the exact value would be $$S_{10}=\frac{43635917056897}{40327580160000}\approx 1.082036583$$

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