Does there exist an alternative way to demonstrate and evaluate the result of the following limit without using an asymptotic expansion? $$\lim_{n\rightarrow+\infty}\dfrac{\displaystyle\sum_{k=1}^{n}\dfrac{1}{k}}{\ln n}$$ Using a spreadsheet presumably the result of the limit is $1$, and watching other posts like  and  related about Euler-Mascheroni constant $\gamma$, the result is obtained immediately. $${\displaystyle \lim_{n\rightarrow+\infty}\dfrac{\displaystyle\sum_{k=1}^{n}\dfrac{1}{k}}{\ln n}=\lim_{n\rightarrow+\infty}\dfrac{\ln n+\gamma+O\begin{pmatrix}\dfrac{1}{n}\end{pmatrix}}{\ln n}}=1$$ Is that correct? However, I would like to know if there is other way to calculate that result using calculus in a context of firsts courses of engineering careers. Maybe with convergence criteria for series?
5047
I'm not completely new to convergent series and sequences, but proving the above gave me some troubles. Would appreciate some help :) Show that: $$\lim_{n\to\infty} \frac{\sum\limits^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$$
59163
I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and the steps used? Thanks. (I prefer to avoid to use L'Hospital's rule.)  $$\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$$
46841
Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.   I showed it using Abel's theorem and limit comparison test. Any other simpler method?
53946
Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$  What method to use in order to find the closed form of summation $\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}$?
3571
Evaluate $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$.   $\bf{My\; Try::}$ Although we can solve it by converting into definite Integration.  But I want to solve it without Using Integration.  So $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} = \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{(n-1)(n-2).......(n-k+1)}{k!\cdot n^{k}\cdot (k+3)}$  Now How can i solve after that, Help required, Thanks
105571
I guess it has something to do with Riemann sums but this is new for me.  $\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$  How do I start?
94944
Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n}$   $\bf{My\; Try::}$we can write it as $$l=\lim_{n\rightarrow \infty}\frac{e^n}{\sqrt{n}}\cdot \left(\frac{1}{n}\cdot \frac{2}{n}\cdot \frac{3}{n}\cdots \cdots \frac{n}{n}\right)$$  $$\ln (l) = \lim_{n\rightarrow \infty}\bigg[n-\frac{1}{2}n+\sum^{n}_{r=1}\ln\left(\frac{r}{n}\right)\bigg]$$  Now how can i solve it, Help required, Thanks
1860
How to show that  $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$  No clue at all. Need help!  Appreciated!
106773
Does $\sum\limits_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$ converge or diverge ? I tried some tests , but nothing conclusive is coming . Pleas help
56291
$$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}}$$   $$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}} =\lim_{n\to \infty} \dfrac1{n}\sum^{n}_{k = 1} \sqrt{\dfrac k n} $$  While searching this question I found, .   Like in the accepted answer I first compared my series to LRAM,    $$\int_a^b f(x)\,dx=\lim_{n\to\infty}\frac 1n\sum_{i=1}^n f\left(a+\frac{b-a}n i \right)$$  I got $a = 0$, $b = 1$ and $f(x) =\sqrt{x}$ so,  $$\int_0^1 \sqrt{x}\ dx = \dfrac2 3$$ should be the answer.  Is there any simpler method to do this sum ? I have not learnt this method to do infinite sums so I can't use it.
62560
$$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number.  Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus teacher told us that it diverges but he doesn't see any elementary way or simple test to prove it. He told us something about Euler's Gamma function and said that he knows it diverges because of Stirling's formula but this is not quite our level yet.  I tried to think about it myself but I established only that this  is some kind of an edge case. We could generalize the problem to series of $\frac{n!a^n}{n^n}$. Then, from the ratio test we know that it converges for $a &lt;e$ and diverges for $a&gt;e$. For $a=e$ the test is inconclusive. I've also tried to compare it with various divergent series, but nothing worked...  Does anybody have any idea for a simple solution?
60890
I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$  I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump after simplifying my limit, though:  $$\lim_{k}\frac{2k^{k}(k+1)}{(k+1)^{k+1}}$$  Normally I'd just derive by L'Hopital's Rule, but the derivative is pretty long and complicated. I doubt that's the correct route.
113177
assume  $$x_n=\frac{n+1}{2^{n+1}}\sum_{k=1}^n\frac{2^k}{k} ,n=1,2,.....$$  how compute $\lim_{n\to +\infty}x_n$?  Thanks for any hint
101257
Could you explain how to find this limit?  $\lim_{n \rightarrow \infty} \sum^n_{k=1} \frac{1}{k+n}$
5154
What is the result of the next limit:   $$ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $$  Why (theorem)?
95733
Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$  I use software to complete the  series is  $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$  I have no idea about it.  :|
24010
Evaluate the limit   $$\lim_{n \to \infty} \dfrac{1}{1+n^2} +\dfrac{2}{2+n^2}+ \ldots+\dfrac{n}{n+n^2}$$   My approach :   If I divide numerator and denominator by $n^2$ I get :   $$\lim_{ n \to \infty} \dfrac{\frac{1}{n^2}}{\frac{1}{n^2} +1} +\dfrac{\frac{2}{n^2}}{\frac{2}{n^2} +1} + \ldots+ \dfrac{\frac{1}{n}}{\frac{1}{n} + 1}=0$$   but the answer is $\dfrac{1}{2}$ please suggest how to solve this.. thanks.