p-series theorem

A p-series, or hyperharmonic series, is a positive-term series (a series Σa_n such that a_n>0 for every n) in the form of:

∞
Σ 1/n ^p = 1 + 1/2 ^p + . . . + 1/n ^p + . . .,
n=1

where p is a positive real number.
(If p = 1, then it is the harmonic series.)

Theorem:

The p-series
∞
Σ 1/n ^p
n=1

1. converges if p > 1
2. diverges if p ≤ 1 (but remember; p must also be greater than zero, or it would be 1/n ^-p = n ^p)

Proof: Using the integral test (all of the conditions are satisfied; f is positive for x ≥ 1 and it is decreasing)

∞
∫ (1/x ^p)dx =
1

      t
lim   ∫ (x(-p))dx = 
(t→∞) 1

(1/1-p)lim(t(1-p) -1).
       (t→∞)

If p > 1, then p - 1 > 0 and the last expression can be written as:

(1/1-p)lim(1/(t(p-1)) - 1) = 
       (t→∞)

(1/1-p)(0-1) = 1/p-1.

Thus, the p-series converges if p > 1. If 0 < p < 1, then 1 - p > 0 and

(1/1-p)lim (t(1-p) - 1) = ∞.
         (t→∞)
Thus, the p-series diverges. If p ≤ 0, then 
lim(1/n p)!= 0 
 (n→∞)
and, by the nth-term test, the series diverges(see the above parenthetic on 1/n-p).

Application for the p-series theorem is to determine if the infinite series (adding each term of the infinite sequence) converges
or diverges. If a series converges, it means that the series has a sum. If a series diverges, it means that the series will add
to infinity, thus, it does not have a sum. An example I like to explain this with is:
consider the series: .3 + .03 + .003 + .0003 ...(to infinity.) This infinite series will go to the sum of 1/3 (.3333333 repeating.)
Although this is not a p-series, it helps to understand adding numbers to infinity. But if this was a p-series, we could have
determined whether it would converge before we found the actual sum (though, the sum of some series is difficult to find).

Examples:

∞
Σ 1/n ² ........ Converges since p = 2 > 1
n=1

∞
Σ 1/√(n) ........ Diverges since p = 1/2 > 1
n=1