Limit of Series: Find Out How (N+1)/(2N+1) = 1/2

[magnifik]

can someone please explain to me how the limit of (N+1)/(2N+1) as N goes to infinity is equal to 1/2? i know this is really simple, but I'm pretty rusty on this stuff.

 

[n.karthick]

If you substitute $$N=\infty$$ you will get undetermined form $$\frac{\infty}{\infty}$$. So you can L' Hospital rule, i.e, differentiate numerator and denominator (individually) with respect to N and then take limit for the resulting expression.

 

[cragar]

When both the top and bottom go to infinity you can look at the coefficients in front of the variable.

 

[spamiam]

When you have a rational function of N, you can divide the numerator and denominator by the highest power of N. In your case:

$$\lim_{N \to \infty} \frac{N+1}{2N+1} =\lim_{N \to \infty} \frac{(1/N)(N+1)}{(1/N)(2N+1)} = \lim_{N \to \infty} \frac{1+1/N}{2 + 1/N} = \frac{1 +0}{2 +0} = \frac{1}{2}$$

since $\lim_{N\to \infty} 1/N = 0$.

 

[Calrid]

It's probably easier to just stick in limits and cancel out the irrelevant details to leave just the numerator and denominator as 1 and 2, slightly shorter process if less robust ie n+1/n+1 in both the numerator and denominator =1 with appropriate limits. That was the way I was taught but the answer above is completely correct. It's intuitively correct to if you can imagine them both converging in an iterative way ie for every value of n the answer is converging to 1/2 in an infinite series.

Sorry nice to see a new way of doing something that is more rigid just pop corn posting in case it gets even more interesting.

 

[magnifik]

thank you all