Question

first series is Summation from n=1 to inf 1/n and another series Summation from n=1to inf 1/n^2, why the first diverges and the second converges whereas both goes to zero

Answer 1

not the both should converge?

Answer 2

going to zero is "necessary but not sufficient"

Answer 3

you can find an easy proof that
\[\sum_{k=1}^n\frac{1}{k}\] can be made arbitrarily large by choosing large enough n

Answer 4

i can give you the idea of the proof, but it is cumbersome to write it all here, easier simply to google it

Answer 5

idea is take
\[\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\] and say that it is less than
\[\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1\] and keep going in this fashion, you can always add 1/2

Answer 6

what should I google ? I mean if i just type the series as I did here then it doesn't lead me to any good website

Answer 7

sorry, "greater than" not :less than

Answer 8

ok

Answer 9

it is called the "harmonic series" and it is a well known divergent series

Answer 10

http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29

Answer 11

so going to zero is certainly not enough, and this is the canonical example of a sequence that goes to zero, where the series does not converge

Answer 12

whereas
\[\sum\frac{1}{k^p}\] does converge if
\[p>1\] but that is p must be strictly greater than 1

Answer 13

integral test will prove this quickest

Answer 14

understood

Answer 15

what would I do without you hehehehe thanks bro

Answer 16

are you mathematician?

Answer 17

I mean you could be an engineer too?

Answer 18

yw
i am just satellite