integral involving greatest integer function

Question

Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n=\int_1^n \frac{x-[x]}{x^2}dx$. Then, what is $S_{10} + T_{10}$?

The only clue that i can get is break the limits of integration according as the value the greatest integer function takes???please solve this for me. The answer i got was $\ln10 + 1$.

Why should we solve your problem? We're happy to try to help you learn, but that is (absolutely) not the same as just doing math for you. — Greg Martin, Feb 10 '16 at 07:30
In fact, "just solve my problem" is a particular kind of "off-topic". — , Feb 10 '16 at 09:59

choco_addicted · Answer 1 · 2016-02-10T06:48:30.973

3

Hint 1:

\begin{align}\int_1^n \frac{x-[x]}{x^2}dx &= \int_1^2 \frac{x-1}{x^2} dx+\int_2^3\frac{x-2}{x^2} dx + \cdots + \int_{n-1}^n\frac{x-n+1}{x^2} dx\end{align}

Hint 2:

\begin{align}\int_{n-1}^n \frac{n-1}{x^2}dx&=\left[\frac{(1-n)}{x}\right]_{n-1}^n\\&=(1-n)\left(\frac{1}{n}-\frac{1}{n-1}\right)\\&=\frac{1}{n}\end{align}

Can you proceed from it?

edited Feb 10 '16 at 06:48

answered Feb 10 '16 at 06:35

choco_addicted

8,906

yes i have done it like this..but still if you could show the working and steps – user312254 Feb 10 '16 at 06:37
I added one more hint. But you have to proof the problem yourself eventually. – choco_addicted Feb 10 '16 at 06:50

integral involving greatest integer function

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