# Thread: How to simplify this question

1. ## How to simplify this question

Determine the series is absolutely convergent or diverges by using ratio test.
b) $\sum ^\infty _{n=1}$ ${(1+\frac{1}{n})}^{n^2}$
$\frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^ {n^2}}$
= $\frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2} }$

${(1+\frac{1}{n+1})}^{2}+{(1+\frac{1}{n+1})}^{2n}$
= $e^{2n} +{e^2}$

2. Originally Posted by anderson
Determine the series is absolutely convergent or diverges by using ratio test.
b) $\sum ^\infty _{n=1}$ ${(1+\frac{1}{n})}^{n^2}$
$\frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^ {n^2}}$
= $\frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2} }$
This does not follow. I think you are arguing that $(x+1)^2= x^2+ 2x+ 1$ so $a^{(x+1)^2}= a^{x^2+ 2x+ 1}= a^{x^2}+ a^{2x}+ a^1$. That last equality is incorrect. That should be a product, not a sum.

${(1+\frac{1}{n+1})}^{2}+{(1+\frac{1}{n+1})}^{2n}$
= $e^{2n} +{e^2}$

3. Dear HallsofIvy,

b) $\sum ^\infty _{n=1}$ ${(1+\frac{1}{n})}^{n^2}$
$\frac {{(1+\frac{1}{n+1})}^{(n^2+2n+1)}}{{(1+\frac{1}{n} )}^{n^2}}$