Linked Questions

7
votes
3answers
620 views

Compact formula for $\sum_k k!$ [duplicate]

Is there any compact formula for: $$\sum_{k=0}^n k!$$ I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$ I've ...
5
votes
1answer
221 views

What is the value of $\sum_{k=1}^{n}k!$? [duplicate]

What is the sum of all the factorials starting from 1 to n? Is there any generalized formula for such summation?
1
vote
3answers
275 views

How to find a closed form of this simple factorial sequence [duplicate]

$$S_1=1\\ S_n=n!+S_{n-1}$$ Is there a simple way to express $S_n$ without summing up all the previous terms? Sorry I haven't put any effort in the problem but I don't know where to start. So this ...
0
votes
2answers
113 views

Is there any formula for $\sum_{k=1}^n k!$? [duplicate]

Do we have any formula for the sum of factorials above?
2
votes
0answers
160 views

Is there any simplification of $1!+2!+3!+4!…n!$? [duplicate]

I am aware that there is a formula $^nC_0 + {^n}C_1 + {^n}C_2 + {^n}C_3 + ... {^n}C_n=2^n$. Is there any similar formula to calculate the value of $1!+2!+3!+4!...n!$? Can this be otherwise denoted ...
1
vote
0answers
41 views

Is there a general formula to the: sum n! [duplicate]

For example as the: sum n has the general formula >>> n(n+1)/2 Is there a formula for the n!
6
votes
2answers
157 views

Prove that $0!+1! + 2! + 3! + … + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three real number

Let $n$, $p$ and $r$ be three positive integers. Prove that for $n \geqslant 3, r>1$, $$\sum_{k = 0}^{n} k! \neq p^\text{r}$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preaparatory Question) I am not so ...
2
votes
4answers
85 views

How fast does $\frac{1}{n!} \sum_{k=0}^n k(n-k)!$ grow, as a function of $n$?

Consider the sum $$\frac{1}{n!} \sum_{k=0}^n k(n-k)!$$ How fast does it grow (as a function of $n$)? I can prove that it grows slower than or equal to linearly (but I expect this is very crude), so I ...
5
votes
1answer
134 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
2
votes
0answers
261 views

The closed form of $\sum_{n=1}^{x}n!$

Let $$y=\sum_{n=1}^{x}n!$$ be the sum of consecutive factorials. What is closed form for $y$ in terms of $x$? Wolfram Alpha says that $$y=-(-1)^x\Gamma(x+2)(!(-x-2))-!(-1)-1$$ where $!x$ is ...
2
votes
0answers
133 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
-3
votes
1answer
80 views

what is remainder from division of A by 10?

What is remainder from division of $A$ by $10$? $A=0!+3!+6!+9!+…+81!$ I thought I was kind of expert in high school mathematics until I was hit in punch by my brother who is in secondary school. Guys ...
0
votes
1answer
80 views

Summation of factorials

Finding the sum of following series $$\frac{1}{1!}+\frac{1}{2!} +...........+\frac{1}{n!}$$ I tried $$\frac{1}{n!}\Bigg[\frac{n!}{1}+\frac{n!}{2!}+\frac{n!}{3!}+..... .....+1\Bigg]$$ And end up ...
0
votes
1answer
40 views

Sum of a series when its not arithmetic

Say you had $$\sum^{1287}_{n=1} (n!)^4$$ How would you got about summing this as it is not arithmetic you cannot use the formula. I tried creating a common difference in terms of $n$ but that did not ...
0
votes
0answers
50 views

What is the general term of $\sum_{i=2}^{n/2} (n-i)!$?

How do you go about solving this summation?: $\sum_{i=2}^{n/2} (n-i)!$ Edit: For context, I am trying to do an analysis of the average case time complexity of merge sort, measured based on the ...