All Questions
21
questions
-1
votes
2
answers
168
views
$\sum_{k=1}^n(k!)(k^2+k+1)$ for $n=1,2,3...$ and obtain an expression in terms of $n$
Find a closed expression in terms of $n$.
$$\sum_{k=1}^n(k!)(k^2+k+1); n=1,2,3...$$
Any idea about how to do this.. I'm a new to this so a little explanation would be helpful. Thanks in advance!
- 127
1
vote
2
answers
136
views
Question about Binomial Sums [duplicate]
Prove that for any $a \in \mathbb{R}$
$$\sum_{k=0}^n (-1)^{k}\binom{n}{k}(a-k)^{n}=n!$$
I rewrote the sum as
$$\sum_{k=0}^n \left((-1)^{k}\binom{n}{k} \sum_{i=0}^n (-1)^{i}a^{n-i} k^{i} \right)$$
...
- 5,719
0
votes
1
answer
96
views
Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$
I'm trying to prove that
$$\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}\left[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}\right]$$
for $p,n \geq 2$, $p, q \in \mathbb{N}$.
I'm trying to use induction ...
- 13
5
votes
3
answers
2k
views
Digit in units place of $1!+2!+\cdots+99!$
There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were $3$, $0$, $1$ and $7$. I am absolutely stumped. Any ...
- 8,867
3
votes
1
answer
223
views
Factorial Identity - True or False?
Let $x$ and $y$ be positive integers.
Then, is
\begin{align}
\frac{x^{xy}}{(xy)!} = \sum_{k_1+...+k_x = xy} \frac{1}{(k_1)!...(k_x)!}
\end{align}
true, where $k_1$, ..., $k_x$ are all positive ...
- 680
10
votes
0
answers
201
views
Finding $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$ [duplicate]
How to find :
$$\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$$
- 101