Harmonic Series and p-Series

I’ve always been boggled by this:

Summation from n = 1 to (infinity) of : 1/n^p
= 1/1^p + 1/2^p + 1/3^p + 1/4^p + 1/5^p +… + 1/n^p + …

This series is said to converge if p > 1, but it diverges if p <= 1

at p = 1, look what’s going on here:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 … + 1/n + …

Now the good ol’ intuition starts working here and i’m thinking, i’m adding on successively smaller and smaller numbers. I’m thinking convergence, but sure as $#!t the S.O.B. goes and fails the damn integral test:

Summation from n = 1 to (infinity) of 1/n =

Integral from N to (infinity) (f(x) dx):

So I work it out:

Integral from 1 to (infinity) (1/x dx) =

limit as k → (infinity) Integral from 1 to k (1/x dx) =

limit as k → (infinity) ln(x) evaluated at x = k and subtract ln(x) evaluated at x = 1

= limit as k → (infinity) ln(k) - ln(1)

I want to get that ln out of there so i raise E to both sides of the equation and get:

limit as k → (infinity) k - 1
DAMN! that sucker pretty much equals (infinity) - 1; which is still infinity. Has anyone else found this harmonic series convergence/divergence stuff baffling?? and AND why is it that p = 1 seems to be the magic number, any number greater than one ( even 1.000000000000000000000000001) would cause this series to converge and any number less than 1 (even 0.9999999999999999999999999999) diverges.
Thoughts anyone???

(x/n)^p : p < 1

Let’s be accurate.

That’s the p-series. The specific p-series you’re thinking of is the harmonic series. An important series…

Check this out, maybe this will help show you how it’s divergent and how your intuition was wrong (like many before you)…

First, know that we can factor all whole-number out as constant adders C such that 1 + 2 + 3 + I … = C.
So you can just say C + (1/2) + (1/3) + (1/4) + (1/5)…

Now…watch this…

C + (1/2) + ( (1/3) + (1/4) ) + ( (1/5) + (1/6) + (1/7) + (1/8) ) ....
         a1   + ( a2    +    a3   ) + ( a4 + a5 + a6 + a7 ) (sigma(a7...a15)) ...
        p1         +      p2            +         p3          +     p4         ....

Now, as we see, p4 > p3 > p2 > p1… so if you add them, they diverge. Hence, the harmonic series is divergent.

However, the series (1/(n^2)) converges… observe…

C + (1/2) + ( (1/4) + (1/8) ) + ( (1/16) + (1/32) + (1/64) + (1/128) ) ....
         a1   + ( a2    +    a3   ) + ( a4 + a5 + a6 + a7 ) (sigma(a7...a15)) ...
        p1         +      p2            +         p3          +     p4         ....

As you see here, p1 < p2 < p3 < p4…

So this converges because as you sum the terms, they are bounded by an upper limit… which comes from the first term.

This is true for p = 1.5 as well. Same dealio as p = 2.

The reason that p = 1 is the upper noninclusive limit is a pain to prove. Just accept it or look it up. But anything greater than p will make the series convergent.

Your assumption that the harmonic series is convergent is incorrect. The SEQUENCE converges, but the SERIES diverges. Know the difference.

Let’s say the sequence T converges to 14. Let’s say you start at T5 instead of T1…

What’s the new limit of the sequence?

Believe it or not, it’s still 14…the terms have no relation between each other. They’re just a sequence of discrete points…therefore all their terms will have the same limit based on the function.

What if T was a series, instead. Let’s say sigma(T, end = infinity, n = 1) = 14. What would sigma(T, end = 9, n = 1) be?

Lim(T, end = infinity , n = 1) - Lim(T, end = 9, n = 1)

If you know this, you know 70% of sequences and series.
PS… 0.99999999999… and 1 are the same number.

Why? What is 1/3rd of 0.99999999…? 0.33333333…

What is one third of 1? 0.3333333333…

What is three times 0.33333333…? 0.999999999…

What is three times 1/3rd? 3/3?

What does 3/3 equal? ONE.

0.99999999999… and 1 are conventionally the same number.

God…I love this stuff.

I love it too. That’s the Bernouli thing. The first time i saw it I was amazed, it makes things so clear. A similar idea I liked was Gauss’ method for computing the series 1+2+3+…+99+100. You guys heard of it? Not real complicated, but the idea can be applied to a lot of other things, it’s basically the same thing as rafajafar is doing here.

YES! I love the Gauss… My discrete math teacher said he did that at age 8 or something crazy young like that… He got in trouble and his teacher asked him to calculate the sum of every number up to 100 before he could go to recess…soooo he did this:

1    + 2     + 3   + 4   + 5   ...... + 96 + 97 + 98  + 99  + 101
100 + 99  + 98 + 97 +  96  ...... + 5   +  4  + 3   + 2    + 1
-------------------------------------------------------------------------
101 +  101 +  101 + 101 + 101 ......  + 101 +  101 + 101 + 101 + 101
100 x 101 = 10100
But this is 2 times the correct answer b/c it's adding once up and once down to each other, so just...divide by two!
5050!!

Rawr, I was like “oh em gee” it’s so easy…and obvious…why didnt I think of that?? LOL.

Rafa, you gotta teach me this shit one day, man. Because I was under the assumption that…

276(>c6)- (n=13*), G7/ 2+2- (P: X>G7)

…only concludes the sequence if snakes eat ice cream.

Am I wrong?

[shoves Rafa into a think tank and locks the door]

How about 10^x = cabin, for x.

Sure, why not?

Its all greek to me.

Wait a minute. Did I accidentally get this right?

“276(>c6)- (n=13*), G7/ 2+2- (P: X>G7)”

[laughing]