## anonymous 4 years ago What is the difference between divergence and convergence?

1. anonymous

I can give you an example to explain if necessary

2. amistre64

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3. amistre64

divergence goes off into infinity; and convergence settles down to something that we can define

4. amistre64

determining "d" or "c" can be a pain tho

5. anonymous

wait just gotta show u something

6. anonymous

$\int\limits_{1}^{\infty}1/x ^{2}$

7. amistre64

convergent I believe; since it gets real small real quick

8. anonymous

and lets say $\int\limits_{1}^{\infty}1/\sqrt{x}$

9. amistre64

x^(-1/2) gets big

10. amistre64

i think i got that right lol

11. anonymous

ya u did lol

12. anonymous

but they are both getting smaller

13. amistre64

1/x is a dividing line; 1/x diverges, but anything just shy of it bigger, like x^(-1.00000000001) converges

14. amistre64

the sqrt doesnt do it fast enough

15. amistre64

1/x goes to gets smaller buth the sums never settle down

16. anonymous

oh ok

17. amistre64

the convergence is called a least upper boundary, not that that makes much difference. It just says that we can add up the partial sums and they tend towards a nice number or they dont

18. anonymous

oh ok get it now thanks :D

19. amistre64

youre welcome :)

20. anonymous

$\int_a^{\infty} \frac{1}{x^r}dx=\int_a^{\infty}x^{-r}dx$ converges if $r>1$ diverges if $r\leq 1$