+646 Let
\(N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). \)
Find N.
∑[(k*(ln(k)/0.34657359)) - (ln(k)/0.34657359)], for k=1 to 1000
N≈9.23734613867.....×10^6
Note: I used natural log(ln) instead of common log(base 10), but it should not alter the result.
Also I converted log(sqrt(2)) to a constant of 0.34657359. Check my result by summing up 3-4 terms using the notation above. I did so and it seems OK.
I may have misread your question!! I multiplied the constant k, by the first term only. Looking at it more closely, it appears that it should be multiplied by the whole expression as follows:
∑k*[(ln(k)/0.34657359) - (ln(k)/0.34657359)], for k=1 to 1000
If this is accurate, then N = 0 !!!.
