Question

What's a Convergent series basically..?? Our professor gave us the idea that if the nth term and its neighberhood terms tends to be a finite(non fluctuating) quantity. But on the internet I am getting answers as divergent even if the nth term tends to zero.. since, then the method for checking it is calculating the sum of the series.

eg.

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...+ 1/n

in this wikipedia claims its divergent.. but according to our professor it should be convergent

anyone please help me out.. i am not sure what is the concept of it!!

Answer 1

The series you mentioned here (harmonic series) is divergent, because the sum of it tends to be infinite. When a series is convergent it converges to a finite quantity.

Answer 2

http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Comparison_test

Answer 3

A convergent series is a series whose infinite sum adds up to a finite number.

Answer 4

thanx a lot .. but i mean what should be the factor for determining whether its convergent or not.. is it only summation tending to a finite quantity or nth term and its neighborhood terms tending to a finite quantity i.e limit exists at n->infinite and \[a _{n} is finite\]

Answer 5

If \[
\lim_{n\to \infty}a_n\neq 0
\]Then we know it is divergent, but we don't know if it is convergent.

Answer 6

if \[\lim_{n \rightarrow \infty} a _{n}= 5\] or any finite quantity is the series divergent or convergent?

Answer 7

If it is greater than \(0\) then it is divergent.

Answer 8

i had the misconception that if limit exists its convergent.. thanx a lot for clearing that up

Answer 9

Remember that \(c\times \infty=\infty\) when \(c>0\).

Answer 10

However when \(c=0\) you get an indeterminate form. It could result in something finite or something infinite.

Answer 11

yep

and so if a series is not convergent it must be divergent?

Answer 12

The \(\times \infty\) comes from the fact that you're adding infinite terms

Answer 13

Yes.

Answer 14

so \[\lim_{n \rightarrow \infty} a _{n } =0\] it should be convergent but the harmonic series i asked in the ques is divergent...

Answer 15

When did I say that means it is convergent?

Answer 16

Okay what is \[
0\times \infty \to?
\]

Answer 17

Does it go to \(\infty\) or to some finite number?

Answer 18

it could be anything

Answer 19

0 to infinite right ..??

Answer 20

If you understand that indeterminate forms means "inconclusive" then why are you struggling with this limit being "inconclusive" when it goes to 0?

Answer 21

ohk so more generally i should go for checking summation of a series and see if its finite .. to conclude its convergence..?

Answer 22

i mean summation of a sequence not series

Answer 23

There are many tests.

Answer 24

You can compare it to another series which you know is convergent or not.

Answer 25

@wio ohk thanx a lot .. i really appreciate you helping me out ..i'll first consult my notes if problem still persists i'll message you..

bye