Question

what is convergence sequence and what is divergence series tell me in easy defination plzzz

Answer 1

there is no royal road to geometry. you have to work from the definition

Answer 2

also please note that you do not use "sequence" and "series" interchangeably. they are different animals

Answer 3

ok tell me in case of series

Answer 4

A divergent series is something that will eventually go to infinity. Example: 1 + 2 + 3 + 4 + 5 + ...
if you just keep adding the numbers, they won't converge.

The terms in the a convergent series eventually decrease enough that they actually equal a single constant number. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

Answer 5

u say "they would not converge" what does it mean

Answer 6

the definition of a convergent series is that the SEQUENCE of partial sums converge. for example, in the example above by ine
the partial sums are
1/2, 3/4, 7,8, 15/16, ... and those converge to 1

Answer 7

does not converge either means the sum does not converge to a single number. this might be because it is infinitely large, or it might simply be because it doesn't approach a single limit

Answer 8

what is partial sums

Answer 9

for example
1 + 1 + 1 + 1 + ... is infinite
whereas
\[1-1+1-1+1-1+...\] has not limit

Answer 10

partial sum
\[Sn=a_1+a_2+...+a_n\]

Answer 11

sum of the first n terms

Answer 12

thanks a lot

Answer 13

@satellite73, it's true that the sum doesn't necessarily converge to a single number; I should have clarified that in my wording. In what sense are you using the term "infinitely large"? do you mean "arbitrarily large"?

Answer 14

yes i guess "arbitrarily large" would be better wouldn't it?

Answer 15

ine is it necessary 4 a converjent series to decrease eventually????

Answer 16

Yes. You might try using the series 1 - 1 + 1 -1 + 1 - ... as a counterexample as it is not divergent, but it is not convergent either.

Answer 17

Basically, if a series converges, the terms of the series must approach 0. *however* - just because the terms of a series approach 0 does not mean it will converge (example 1 + 1/2 + 1/3 + 1/4 + ..., the harmonic series, does not converge.

Answer 18

@satellite73 - checking up on my terminology, it appears that at least Wikipedia is willing to vindicate me. They consider the Grandi series to be divergent ( http://en.wikipedia.org/wiki/Grandi%27s_series)

Answer 19

how can u say that 1+1/2+1/3+1/4...... not converges