Question

determine whether the series converges or diverges ((-1)^n)/n ln(1 + 1/n)

Answer 1

\[\frac{ (-1)^n }{ n }\ln(1+\frac{ 1 }{ n })\]

Answer 2

to get peoples attention you can use @ then type their username or type one letter and some usernames that are online pop up that will notfiy them

Answer 3

\[\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\ln\left(1+\frac{1}{n}\right)\]
Have you tried the alternating series test?

Show that the sequence is monotonically decreasing and \[\lim_{n\to\infty}\frac{1}{n}\ln\left(1+\frac{1}{n}\right)=0\]

Answer 4

i got that it converges. Finding if it monotonically decreases is what's giving me the most problems. i know that bn>b n+1. i can't seem to find the value of n at which the whole thing is decreasing

Answer 5

Do you have to explicitly show the value for which it starts decreasing?

Answer 6

yep

Answer 7

Have you tried the derivative test? You find the critical point of the function
\[f(x)=\frac{1}{x}\ln\left(1+\frac{1}{x}\right)\]

Answer 8

it's just the first derivative right?

Answer 9

Yes

Answer 10

yeah, it got complicated

Answer 11

\[\begin{align*}f'(x)&=-\frac{1}{x^2}\ln\left(1+\frac{1}{x}\right)+\frac{1}{x}\left(\frac{-\frac{1}{x^2}}{1+\frac{1}{x}}\right)\\
&=-\frac{1}{x^2}\ln\left(1+\frac{1}{x}\right)-\frac{1}{x^2(x+1)}\\
&=-\frac{1}{x^2}\left(\ln\left(1+\frac{1}{x}\right)+\frac{1}{x+1}\right)
\end{align*}\]
Does that help? Set it equal to 0 and solve for x.

Answer 12

i was afraid it'd look like that. but thanks!