Vrftsjtryk

$$x_n = 1 + 1/2 +dots +1/n- log n$$

Then -

$1.$ Is the sequence increasing?

$2.$ is the sequence convergent?

For $(1)$, $sum 1/n$ is increasing and $log n $ is also increasing. First few terms are increasing, but i don't know about later terms.

$(2)$ $n^{th}$ term of the sequence can be written as $a_n = (sum_{i=1}^{n}) - log n$

So, $lim_{nto infty} a_n = lim_{n to infty} sum 1/n -lim_{n to infty} log n$

Neither first part nor second is convergent here. so i could not conclude anything.

How to solve?

This sequences converges to the Euler–Mascheroni constant.
It’s very important in number theory.

Let $x_n=-log(n)+sum_{k=1}^n frac1k$. Then, using $log(1+x)ge frac{x}{1+x}$, we see that

$$begin{align}
x_{n+1}-x_n&=frac1{n+1}-logleft(1+frac1nright)\\
&le frac1{n+1}-frac{1}{n+1}\\
&=0
end{align}$$

and $x_n$ is decreasing.

Next, we can estimate the harmonic sum as $sum_{k=1}^n frac1kge frac12 sum_{k=1}^{n-1}left(frac1k+frac1{k+1}right)$, which represents the Trapezoidal Rule approximation of $int_1^n frac1x,dx$.

Inasmuch as $frac1x$ is convex, the trapezoidal rule approximation overestimates the integral of $frac1x$ and we have

$$sum_{k=1}^nfrac1k-log(n)ge sum_{k=1}^n frac1k -log(n)-frac12-frac1{2n}ge0 $$

whence we see that

$$x_nge frac12$$

Since $x_n$ is decreasing and bounded below by $frac12$, the sequence $x_n$ converges.