Infinite series and partial sums: sum[n=1,infty] 1/n^{19/17}

[Kemikeren]

I have the following problem:

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Consider the infinite series

. . . . .\(\displaystyle \displaystyle S\, =\, \sum_{n=1}^{\infty}\, \dfrac{1}{n^{19/17}}\)

and the partial sum

. . . . .\(\displaystyle \displaystyle S_N\, =\, \sum_{n=1}^N\, \dfrac{1}{n^{19/17}}\)

Determine N such that

. . . . .\(\displaystyle N\, \mbox{sa}\, \big|\, S\, -\, S_n\,\big|\, \leq\, \dfrac{17}{2}\, \times\, 10^{-84}\)

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I know that the error is the first "neglected" part of the sum i.e. if I sum from n=1 to n=4 then the |S-S_N| ≤ n=5 (This might be a bad explanation, but I don't know how to use math mode on this forum yet.

Anyway. I assume that I have to find the value of N so that my sum goes up until the value before 17/2*10^(-84) but how do I do this?

Thank you very much in advance!

(And please, if you know how to insert equations in this forum, please let me know )

 

[stapel]

I have the following problem:

---

Consider the infinite series

. . . . .\(\displaystyle \displaystyle S\, =\, \sum_{n=1}^{\infty}\, \dfrac{1}{n^{19/17}}\)

and the partial sum

. . . . .\(\displaystyle \displaystyle S_N\, =\, \sum_{n=1}^N\, \dfrac{1}{n^{19/17}}\)

Determine N such that

. . . . .\(\displaystyle N\, \mbox{sa}\, \big|\, S\, -\, S_n\,\big|\, \leq\, \dfrac{17}{2}\, \times\, 10^{-84}\)

---

I know that the error is the first "neglected" part of the sum i.e. if I sum from n=1 to n=4 then the |S-S_N| ≤ n=5 (This might be a bad explanation, but I don't know how to use math mode on this forum yet.

Anyway. I assume that I have to find the value of N so that my sum goes up until the value before 17/2*10^(-84) but how do I do this?

Thank you very much in advance!

(And please, if you know how to insert equations in this forum, please let me know )

I've converted the (fairly small) image into mathematical formatting. You can do the same by using the LaTeX tool that this forum accepts (here).

I was unable to convert one part of the image, because I don't know what it means. What is the "sa" in the last line of the exercise? Thank you!