## agentx5 3 years ago True or False & why? (n+1)! = n!(n+1) This question is relating to series.

1. 91

yes

2. ParthKohli

Woah. You confused on here?

3. ParthKohli

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

4. ParthKohli

That's an axiom, isn't it?

5. agentx5

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

6. ParthKohli

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

7. ParthKohli

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

8. agentx5

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

9. vishweshshrimali5

$n! = n (n-1)(n-2)...$ SO, $(n+1)! = (n+1)n(n-1)(n-2)... = (n+1)n!$

10. agentx5

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

11. ParthKohli

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

12. agentx5

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

13. ParthKohli

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

14. vishweshshrimali5

$THANKS$ @ParthKohli

15. vishweshshrimali5

$Correct$

16. ParthKohli

$\prod_{i = 1}^{n}i = n!$

17. Neemo

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

18. agentx5

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

19. vishweshshrimali5

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

20. ParthKohli

It's actually a simple fact.

21. agentx5

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

22. ParthKohli

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

23. ParthKohli

Let's just express 3! in terms of 2!. $$3! = 3 \times 2 \times 1 = 3 \times (2 \times 1) = 3 \times 2!$$

24. Neemo

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

25. experimentX

|dw:1343146818151:dw|

26. experimentX

|dw:1343146998874:dw|

27. ParthKohli

Woah! That's some hardcore Mathematics!

28. agentx5

Yeah I'm getting an interval of convergence of -0.5 < x < 0.5 And a radius of convergence of 0.5

29. agentx5

Would you agree with that @experimentX ? The limit for this series goes to infinity, it diverges.

30. experimentX

wait .. something went wrong!!

31. agentx5

I'm using the ratio test... I think that's what you did too *looks back at what your wrote*

32. ParthKohli

What a change! $$\mathbf{Factorials \Longrightarrow Calculus}$$

33. experimentX

well .. factorials are the basics!!

34. experimentX

lol ... i did the opposite!!

35. agentx5

$\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right|$ $\lim_{n \rightarrow \infty} \left| \frac{(n+1)!(2x-1)^{n+1}}{n!(2x+1)^n} \right|$

36. agentx5

Using the axiom Parth pointed out I can then cancel...

37. agentx5

$\lim_{n \rightarrow \infty} \left| \frac{\cancel{n!}(n+1)(2x-1)^{\cancel{n}+1}}{\cancel{n!}\cancel{(2x+1)^n}} \right| = \infty$

38. agentx5

Divergent by the Ratio Test @ParthKohli & @experimentX if 2x-1 = 0 then x $$\neq$$ 0.5 So the interval of convergence is only from -0.5 to 0.5, non-inclusive

39. agentx5

But the radius? 0.5? 0? o_O

40. ParthKohli

Oh no! Not THAT type of limits!

41. agentx5

PS: People please give @experimentX a medal...

42. ParthKohli

Done.

43. experimentX

|dw:1343147306986:dw| nvm .. i'm at 99, can't grow any further. i guess ratio test is bad test for this series!!

44. experimentX

|dw:1343147733699:dw|

45. agentx5

Hold on ladies & gents, let me re-ask this specific problem as a new question, that way proper credit can be due and we're not all off-topic technically :-D

46. experimentX

no ... i enjoy weird problems!!