LImit with and exponent variable

**jman919** · February 26th 2011, 03:02 PM

What is the limit of the sequence

LImit with and exponent variable-codecogseqn.gif

as n approaches infinity.

**emakarov** · February 26th 2011, 03:15 PM

Hint: $a_n=\left(1+\frac{2}{n+1}\right)^{n+1}/\left(1+\frac{2}{n+1}\right)$ .

**jman919** · February 26th 2011, 05:06 PM

Ok i understand how you got that.. but I'm not sure where to start. I keep getting that the limit is 1 which I know it is not.

**NOX Andrew** · February 26th 2011, 05:19 PM

Here is a useful equation:

$\displaystyle \lim_{x\to\infty} \left ( 1 + \dfrac{r}{x} \right ) ^{x} = e^r$ .

For your problem, instead of evaluating the limit of the ratio, evaluate the ratio of the limits:

$\displaystyle \lim_{n\to\infty} \dfrac{ \left ( 1 + \dfrac{2}{n+1} \right ) ^{n+1} }{1 + \dfrac{2}{n+1}} = \dfrac{ \displaystyle \lim_{n\to\infty} \left ( 1 + \dfrac{2}{n+1} \right ) ^{n+1} }{ \displaystyle \lim_{n\to\infty} \left ( {1 + \dfrac{2}{n+1}} \right ) }$

The limit in the denominator is 1. Division by 1 won't change anything, so the answer to your original problem is just the limit in the numerator.

$\displaystyle \lim_{n\to\infty} \left ( 1 + \dfrac{2}{n+1} \right ) ^{n+1}$

If we let r = 2 and x = n + 1, then the limit in your problem becomes $\displaystyle \lim_{x\to\infty} \left ( 1 + \dfrac{r}{x} \right ) ^{x}$ . Remember the useful equation at the top?

The limit is just e^r. Substituting r = 2 gives: e^2.

**jman919** · February 26th 2011, 06:34 PM

Thank you so much , it makes sense now.

My mistake was not evaluating the numerator correctly. I found the limit of the denominator correctly, but for the numerator I said that the limit when n approaches infinity was = (1)^ infinity which is the same as 1. What was wrong with this reasoning?

**NOX Andrew** · February 26th 2011, 06:43 PM

$1^\infty$ is what is called an indeterminate form. It's value cannot be determined. In this problem, it turns out it is "equal" to e^2. In another problem, it might turn out to be "equal" to sin(1). Some other common indeterminate forms are $\frac{0}{0}$ and $\infty - \infty$ . Here is a link to a Wolfram article on indeterminate forms: Indeterminate -- from Wolfram MathWorld

**jman919** · February 26th 2011, 06:49 PM

Oh , thats right. I completely forgot that it was indeterminate.

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