I'm trying to figure out the closed form formula for the following:  $$ \sum_{i=1}^{N} \frac{1}{i} $$  I think the denominator will have $N!$ in it but otherwise I cannot figure it out. Can someone help?
91848
Is there any formula for this series?     $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
76270
If we knew $\sum_{i=1}^{n} f(i)=S$  ($n$ can be $\infty$), is there anything we could say about $\sum_{i=1}^{n} \dfrac {1}{f(i)}$ in terms of $S$?  The only thing I was able to find on the web is . If there is no known relationship, is it possible for there to exist one that we just haven't discovered?
21730
Is there a closed formula for:  $\sum_{i=1}^{N}{\binom{i+k}{i}}$   ( k is a constant whole number )
6668
How can I find the formula for the following equation?  $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$$  More importantly, how would you approach finding the formula? I have found that every time, the denominator number seems to go up by $n+2$, but that's about as far as I have been able to get:  $\frac12 + \frac16 + \frac1{12} + \frac1{20} + \frac1{30}...$ the denominator increases by $4,6,8,10,12,\ldots$ etc.  So how should I approach finding the formula? Thanks!
25555
I'm finding a closed form to  $\sum^{n}_{k=1} \frac{k}{(k+1)!}  (n \leq 1) $ (in a environment of induction and recurrence)   I've been trying to solve it without success, can anybody help me (?)  The last thing I've tried to do is write the expression as a Non-Homogeneous Linear Recurrence:   $ f(1) = \frac{1}{2} $   but, actually I have no idea of how to attach it.   $ f(n+1) = f(n) + \frac{1}{(n+2)n!} $
30701
$$\sum_{i=1}^n \frac1{n+i} = \sum_{i=1}^{2n}\frac{(-1)^{i+1}}i,\quad \text{ for }n\ge 1.$$  I really have no clue how on to solve this one. It's the last one on my assignment. Would really enjoy some help with this one.
49817
Lets define the following series $$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$ I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc.. the numerator in all results is 1the pattern in denominator is $[1,4,1,96,600...]$ and   that is equal to $n*n!$ Finally I want to prove the general equality: $\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}=\frac 1 {n*n!}$
9714
Find this sum $$\sum_{n=1}^{\infty}\dfrac{1}{n!(n^4+n^2+1)}$$  My try: since $$n^4+n^2+1=(n^2+1)^2-n^2=(n^2+n+1)(n^2-n+1)$$ then $$\sum_{n=1}^{\infty}\dfrac{1}{n!(n^4+n^2+1)}=\sum_{n=1}^{\infty}\dfrac{1}{2n\cdot n!}\left(\dfrac{1}{n(n-1)+1}-\dfrac{1}{n(n+1)+1}\right)$$ then I can't go further.
67934
$$\frac{\sum_{i=0}^nx_iy_i-\bar y\sum_{i=0}^nx_i}{\sum_{i=0}^nx_i^2-\bar x\sum_{i=0}^nx_i}= \frac{\sum_{i=0}^n(x_i-\bar x)(y_i-\bar y)}{\sum_{i=0}^n(x_i-\bar x)^2}$$  Maybe it refers some calculation rules about summation. I cannot solve that. Thanks!
37354
$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
94783
How can I prove that $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$? I noticed that in the sum, the denominator has terms that cancel out, but I'm not sure how to take advantage of that.
30374
I have to calculate sum of series $\sum \frac{n^2}{n!}$. I know that $\sum \frac{1}{n!}=e$ but I dont know how can I use that fact here..
65961
I was trying to simply an expression in an exercise related to randomized algorithms. Here is the expression which I have obtained at the end.  $$ \displaystyle\frac{\displaystyle\sum_{i=1}^{n+k-1} i \binom{n-i}{k-1}}{ \displaystyle{n \choose k}}$$  Is there any way to simplify the numerator so that the whole expression simplifies into a nice closed formula? A combinatorial approach would be greatly appreciated.
54720
Mathematica tells me that  $$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$  Although I have not been able to come up with a proof.  Proofs, hints, or references are all welcome.
102070
I've worked out a few summation formulas, and I am hoping to find a pattern. Unless I have made a mistake somewhere, we have the following identities:  $$\sum_{1 \le i \le n} i = \frac{(n+1)n}{2} $$  $$\sum_{1 \le i &lt; j \le n} ij = \frac{(3n+2)(n+1) \, n \, ( n-1)}{24}$$   $$\sum_{1 \le i &lt; j &lt; k \le n} ijk = \frac{(n+1)^2 \, n^2 \, (n-1)(n-2) }{48}$$   It seems clear that $$p_k(n)= \sum_{1 \le i_1 &lt; i_2 &lt; \dots &lt; i_k \le n} i_1 i_2 \cdots i_k $$ is a polynomial in $n$ of degree $2k$. But is there a nice closed-form formula for it? Maybe in terms of its factorization?  Comment: I realize that what I am looking for is the coefficient of $t^k$ in the expansion $$(1+t)(1+2t) \dots (1+nt),$$ so maybe generating function techniques could be helpful. But the main way I know to extract the coefficient of $t^k$ is to take derivatives $k-1$ times and on the surface of things that looks like a huge mess.
89276
Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
68775
Who knows about the summation of this series: $$\sum\limits_{i=1}^{n}i^m $$  where $m$ is constant and $m\in \mathbb{N}$?  thanks