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For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$

Find $\sum_{n = - \infty}^{\infty} f(n)$.

I do not know how to solve such problem of series. So I could not try anything.

Thank you for your help.

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  • $\begingroup$ This question seems very much familiar to me. where I have seen this question? asked in some exam? $\endgroup$
    – GA316
    Nov 9, 2013 at 19:18
  • $\begingroup$ NBHM 2006 Prob. 5.2 $\endgroup$
    – Supriyo
    Nov 10, 2013 at 0:17

1 Answer 1

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HINT:

Using Fermat's Little theorem, $$n^p-n\equiv0\pmod p$$ where $p$ is any prime and $n$ is any integer

$\displaystyle\implies n^7-n\equiv0\pmod 7\implies \frac{n^7-n}7$ is an integer

Show that $k(n)$ is integer for all integer $n$

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  • $\begingroup$ Thank you Lab. I made a mistake thinking it as an analysis question. Similarly $\frac{n^3 - n}{3}$ is also an integer. So $k(n)$ is always an integer, and the sum is $0$. $\endgroup$
    – Supriyo
    Nov 10, 2013 at 0:26
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    $\begingroup$ @Samprity, my pleasure. this can be established using induction as well, related : math.stackexchange.com/questions/443538/… $\endgroup$
    – lab bhattacharjee
    Nov 10, 2013 at 9:23

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