Question

Prove that the series [ln((n+1)/n)] from n=1 to infinity is divergent using the fact that ln((n+1)/n) = ln(n+1)-ln(n) and write out a few terms.

Answer 1

Did you try to write out some terms?

Answer 2

Yup. I got {0.7, 0.4, 0.4, 0.2...}

Answer 3

I recommend not simplifying...

Answer 4

like use the hint given

Answer 5

Should I use one of the tests? I was just thrown off by the info given.

Answer 6

write out the first few terms of the series

Answer 7

\[\lim_{n \rightarrow \infty } \sum_{i=1}^{n} (\ln(n+1)-\ln(n))\]
begin by replacing n with 1
then n with 2
then n with 3
then n with 4
...
You should see a pattern


Do not use a calculator

Answer 8

It looks like it's converging toward zero. Is that off-base?

Answer 9

All I'm asking you to do is do this:
ln(1+1)-ln(1)
+
ln(2+1)-ln(2)
+
ln(3+1)-ln(3)
+
ln(4+1)-ln(4)
+
ln(5+1)-ln(5)
+
ln(6+1)-ln(6)

what do you notice?

Answer 10

Oh! I see that both parts are increasing!

Answer 11

:(

that isn't what I'm trying to get you to see

Answer 12

look ln(2)-ln(2)=?
ln(3)-ln(3)=?
ln(4)-ln(4)=?

Answer 13

Oh! I didn't look at it as a whole. It's all actually adding up to zero!

Answer 14

well except for one term

Answer 15

what term is left over?

Answer 16

The final term in the sequence, right? Ln(infinity)? :/

Answer 17

do you think ln(n+1) or ln(n) is what is left over?