Can anyone help me with the following problem please:
Using convergence tests, determine whether the series
∑ (-1)n/√(n+5) over n=1 and n=∞
converges or diverges.
Many thanks to anyone who can provide me with any advice on solving this problem.
Can anyone help me with the following problem please:
Using convergence tests, determine whether the series
∑ (-1)n/√(n+5) over n=1 and n=∞
converges or diverges.
Many thanks to anyone who can provide me with any advice on solving this problem.
Hi
$\displaystyle \sum_{n=1}^{\infty}(-1)^n\frac{1}{\sqrt{n+5}}$
This series is alternating hence you can try to show the convergence using the alternating series test.
Be careful with that n not k...
$\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}}{\sqrt{n+5}}$
Expanding upon Flying squirrels explanation...since $\displaystyle a_{n+1}<a_n$
Where $\displaystyle a_n=\frac{1}{\sqrt{n+5}}$
and also since $\displaystyle \lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{1}{\sq rt{n+5}}=0$ we can see by the alternating series test that this series is at least conditionally convergent
That is not what you had in your post you had that the $\displaystyle \lim_{n \to \infty}|a_n|=0$
But neither of the conditions:
$\displaystyle \lim_{n \to \infty}|a_n|=0$
or
$\displaystyle \exists N;\ \forall n>N,\ |a_{n+1}| \le |a_n|$
on its own is enough you need both.
RonL
if you look closely I had both....I didnt state that n>0 but I had that condition
and I took $\displaystyle \lim_{n\to\infty}\frac{1}{\sqrt{n+5}}=\lim_{n\to\i nfty}\bigg|\frac{(-1)^{n}}{\sqrt{n+5}}\bigg|$
are you saying that my prolem is that I just did it and did not explicity state what I was doing?