Sum from 1 to infinity of:
n/(1000n + 1) (does this converge or diverge...why?)
My thought is take the lim from 1 to infinity of the above expression.
That will end up 1/1000. So this converges...Right?
$\displaystyle \sum_{n=1}^{\infty}\frac{n}{1000(n)+1}$
Note: The divergence test states if
$\displaystyle \lim_{n\to \infty} a_n \ne 0$
If the limit of the series is not 0, the series diverges, you can simply take limit here and apply L Hoptials rule since the limit is of indeterminate form here, and notice the limit here is not 0, so it diverges, or you can do as acc100jt did, I know when i have a rational fraction with equivalent degree in denominator and numerator the limit is the ratio of the coefficients. However, but are ways of showing how the limit is not 0.
$\displaystyle a_{n}=\frac{n}{1000n+1}$
and $\displaystyle \lim_{n\rightarrow\infty} a_{n}=\lim_{n\rightarrow\infty} \frac{n}{1000n+1}=\lim_{n\rightarrow\infty}\frac{1 }{1000+\frac{1}{n}}=\frac{1}{1000}$.
By Test for Divergence, the series $\displaystyle \sum_{n=1}^{\infty}\frac{n}{1000n+1}$ is divergent