Articles of summation

What is the infinite sum of logarithm lnK divided by k(K+1)?

I was trying to calculate the following integral: $\displaystyle\int_1^\infty\frac{dx}{x \lfloor x \rfloor}=? $ which I found to be equivalent to $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k(k+1)} $. There is a close relative of this sum which is $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k^2}=-\zeta^\prime(2)$ and its value is known in terms of Glashier-Kinkelin constant (A): $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k^2}= \frac{\pi^2}{6}[12\ln(A)-\gamma-\ln(2\pi)]$ My idea for solving this was to assume a […]

Sum of first $n$ triangle numbers, without induction

Background I wish to calculate $$ S= \sum_{i = 1}^{n}\frac{k(k+1)}{2}$$ I know what the answer is going to be, since this is essentially the sum of the first $n$ triangle numbers. I.e. $S = (1) + (1+2) + (1+2+3) + \cdots + (1+2+3+\cdots+n)$ All solutions I’ve seen seem to know in advance what the answer […]

simplifying the sum $\sum_{j<k} T_{jk}$ for symmetric $T$

Suppose $T$ is symmetric such that $T_{ab} = T_{ba}$ for all its indices $a,b \in \{ 1, \ldots, n \}$. If I consider the sum: $$ \sum_{a<b} T_{ab} $$ How can I simplify this? I $think$ that this should be equal to the following: $$ \sum_{a<b} T_{ab} = \sum_{b=1}^{n} \sum_{a=1}^{b-1} T_{ab} = \frac{1}{2} \sum_{a,b=1}^{n} T_{ab} […]

Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$

I’m having trouble on starting this induction problem. The question simply reads : prove the following using induction: $$1^{2} + 2^{2} + …… + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + …… + n^{2}$$

Summation of a series

I’m trying to solve a recurrence relation and came across this term $\sum_{i=0}^n i9^i$? I thought this was a geometric series, but I guess it’s not. Is it possible to solve this?

Sum of Gaussian Sequence

I am looking for a closed form or an estimation for the sum of a Gaussian sequence expressed as $$ \sum_{x=0}^{N-1} e^{\frac{-a}{N^2} \: x^2} $$ where $a$ is a constant positive integer. The interesting part is that I have simulated this sequence using MATLAB for $N=0$ to $10^6$ and found that the result is linear […]

A formula for $\int\limits_0^\infty (\frac{x}{e^x-1})^n dx$

The Stirling numbers of the first kind $\begin{bmatrix} n \\ k \end{bmatrix}$ are defined by $\sum\limits_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}x^k:=\prod\limits_{k=0}^{n-1}(x+k)$ with $n\in\mathbb{N}_0$ . Which proof exists for $$\frac{1}{n}\int\limits_0^\infty \left(\frac{x}{e^x-1}\right)^n dx=\sum\limits_{k=1}^n (-1)^{k-1}\begin{bmatrix} n \\ n-k+1 \end{bmatrix}\zeta(k+1)\quad,\quad n\in\mathbb{N}$$ ? I am also looking for literature where a proof is written, so that it isn’t necessary to […]

$\sum_{k=1}^{m+n} \binom {m+n}k k^m (-1)^k=0 , \forall m,n\in \mathbb N$?

Is it true that $\sum_{k=1}^{m+n} \binom {m+n}k k^m (-1)^k=0 , \forall m,n\in \mathbb N$ ? I feel that it is true because if we define $H_1 (x,r)=rx(1+x)^{r-1}$ , and $H_{m+1}(x,r)=x \dfrac d {dx} {H_m (x,r)}$ , then $H_m (x,r)=\sum_{k=1}^{r} \binom {r}k k^m x^k$ and I have noticed , for first few $m$ that we can […]

Find the sum of this series

May I know how I should go about finding the sum of this series? $\displaystyle\sum_{n=1}^\infty$ $\dfrac{n}{2^{n-1}}$ I am really stuck. Thanks!

Proving arithmetic series by induction

How do I prove this statement by the method of induction: $$ \sum_{r=1}^n [d + (r – 1)d] = \frac{n}{2}[2a + (n – 1)d] $$ I know that $d + (r – 1)d$ stands for $u_n$ in an arithmetic series, and the latter statement represents the sum of the series, but I’m not sure how […]