Now I know how this works- but I came across this example and even though I know the answer- the simplification given in the explaination doesn't make sense to me. the squence is a_{n}= {5^{n}/n!} now applying a_{n+1} and dividing a_{n+1}/a_{n} the book indicates = 5/n+1 this is what I don't get how (5^{n+1} /(n+1)!)/(5 ^{n}/n!) can simplify to that ? can someone explain please- what am I missing here.
We have... [tex]\frac{5^{n+1}}{(n+1)!}\frac{n!}{5^n} = \frac{5\cdot5^{n}}{(n+1)n!}\frac{n!}{5^n}[/tex] ...which very easily simplifies to the expression you provided by cancelling out like terms.
right - this is what is not clear to me- I am very new to pure maths how (n+1)! can be written as - (n+1)n!- may be I am having a dumb moment
well n! means = any number say 5 then multiplied by 5x4x3x2x1 ( natural numbers in hughest to lowest order)
So, you have [tex](n+1)!=(n+1)*n*(n-1)*(n-2)*...*3*2*1[/tex] Right? But then we have [tex](n+1)!=(n+1)*[n*(n-1)*(n-2)*...*3*2*1][/tex] And the thing in brackets look familiar, no?? Indeed, the bracketed thing is n! So [tex](n+1)!=(n+1)*n![/tex]
thank you this makes sense- sometimes I just get frustrated with not enough explaination at beiggners level