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November 28

Limit of Infinite Sum and Difference

Given $d(x)=\sin x$ (where x is a natural number) and $D(x)=\sum _{m=1}^{x}\left(d(m)\right)$ , what is the value of $\lim _{x\to \infty }{\frac {1}{x-1}}\sum _{n=1}^{x}\left(d(n)-{\frac {D(n)}{n}}\right)^{2}$ ? This is not for a homework assignment or anything, just asking out of pure curiosity. Thanks for your help. 70.54.112.243 (talk) 05:35, 28 November 2015 (UTC)[reply]

This isn't exactly rigorous but there may be potential to make it so. What you're basically asking for is the average value of (d(n)-D(n)/n)². Since d oscillates between -1 and 1 with no preference for positive or negative, its average value, namely D(n)/n, converges to 0. So the D(n)/n term can be ignored and the problem reduces to finding the average value of (d(n))² = sin² n. But sin² n = 1/2 - 1/2 cos 2n. Arguing as before, the average value of cos 2n is 0 so the average we want is 1/2. --RDBury (talk) 18:11, 28 November 2015 (UTC)[reply]

Great answer, thank you! 70.54.112.243 (talk) 22:36, 28 November 2015 (UTC)[reply]