Question

If we have to argue that a function / sequence is convergent, how would you then use the definition of convergence?

Answer 1

Please use the example \[(\frac{ n }{ n+1 })^n\] to do the arguement

Answer 2

Since it is sequence or function and not a series you can just take the limit as n goes to the infinity.\[\lim_{n \rightarrow \infty} \left( \frac{ n }{ n+1 }\right)^n=\lim_{n \rightarrow \infty} \left(\frac{ n }{ n(1+\frac{ 1 }{ n }) }\right)^n=\lim_{n \rightarrow \infty} \frac{ 1 }{ \left( 1+ \frac{ 1 }{ n } \right)^n}\]And that goes to 1 since all terms have n in the denominator ecept for 1and all of those goes to 0

Answer 3

Oh no forget it, the limit is wrong but that is the idea

Answer 4

The denominator is the definition of the mathematical constant e, so it is 1/e

Answer 5

Okay emm.. well i have done the limit for the function going towards infinity and i get e^-1.
But i ment that i have to use a definition. you know;
A function f from R to R is continuous at a point p∈R if
given ε > 0 there exists δ > 0 such that if |p - x| < δ then |f(p) - f(x)| < ε.

And then i thought that this could solve it for ε=0,1
|(n/(n+1))^n-e^(-1)|<0.1 and n>= 0

Answer 6

That is the definition of a limit, not continuity, the definition for continuity is: f is continuous if the limit at a point has the same value as the function. And you are asking about convergence, wich the definition as I know it is the limit when x goes to infinity.

Answer 7

Ok bad translation from 2 languages to English but here we go, the definition i have been given:
the function f (x) converges to a number a if for every real number ε> 0 (no matter how small), there exists a number N ∈ N such that | f (x)-a | <ε for all x ≥ N.

Answer 8

Yes, that is it, it is the same thing as the limit, isn't it?

Answer 9

It is close but not 100% alike. For converges we are intrested to know something about f(a), for the definition of lim we intrested in f(x) for x close, but different, from a

Answer 10

For the limit you don't need to have x close, and I didn't understand what you said about f(a), isn't a the number to wich f(x) converge? How can a be equal x? And convergence doesn't say anything about a specific point, if that is what you said, it makes sense only if you go to the infinity with n, when you take this limit, x is as big as it can be, and by the definition epsilon would be as small as possible and that is the limit.