Question

True or False & why?

(n+1)! = n!(n+1)

This question is relating to series.

Answer 1

yes

Answer 2

Woah. You confused on here?

Answer 3

\((n + 1)! = n! \times (n + 1)\)

Answer 4

That's an axiom, isn't it?

Answer 5

n! is a factorial though, in this case n will be infinite. I don't know, I'm asking if it's an axiom lol

Answer 6

\( \color{Black}{\Rightarrow (4 + 1)! = 4! \times 5 = 1 \times 2 \times 3 \times 4 \times5 = 5!}\)

Answer 7

That is an axiom, and that's also a way people prove that \(0! = 1\). :)

Answer 8

I suppose that makes sense, it just seems odd to me

Answer 9

\[n! = n (n-1)(n-2)...\]

SO,

\[(n+1)! = (n+1)n(n-1)(n-2)... = (n+1)n!\]

Answer 10

\[\sum_{n=1}^{\infty} n!(2x-1)^n\] for example, is one of the easier ones

Answer 11

\(n! = n(n - 1)(n - 2) \cdots 1\)
Correction* :)

Answer 12

At first I couldn't tell what they were doing to get rid of the n!'s on the ratio test

Answer 13

I believe that a factorial may be expressed as \(\prod\).

Answer 14

\[THANKS\]

@ParthKohli

Answer 15

\[Correct\]

Answer 16

\[\prod_{i = 1}^{n}i = n!\]

Answer 17

sometimes we take 0!=1 as a convention :)

Answer 18

Alright so that's the trick, anything else I should know about factoring n! out?

Answer 19

Well......... Nothing so important.

Answer 20

It's actually a simple fact.

Answer 21

Simple, but not intuitive at first glance, at least not to me

Answer 22

\( \color{Black}{\Rightarrow \Large {16! \over 8!} = {16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \cancel{\times 8!} \over \cancel{8!}}}\)

Answer 23

Let's just express 3! in terms of 2!.

\(3! = 3 \times 2 \times 1 = 3 \times (2 \times 1) = 3 \times 2!\)

Answer 24

a small execise : re-write \[\frac{1.3.5.7.........(2n+1)}{2.4.6.8.............2n}\]
using "!"