Understand why the harmonic series diverges

[touqra]

I understand why the harmonic series, $$\sum_{n=1}^\infty\ 1/n$$ diverges as claimed in math books. What they did was grouping the fractions together and noting that they add up to 1/2, things like that.

But, my logic is this:

$$\lim_{n\rightarrow\infty} 1/n = 0$$

and the elements in the harmonic series are decreasing gradually from 1 until it hits 0.

And, so since, it is decreasing gradually, logically, when you add all the terms together, the series must converge right?

how will harmonic series diverge?

 

[Justin Lazear]

Are you familiar with the integral test for convergence?

Do you know why

$$\lim_{N \to \infty} \int_1^N \frac{dn}{n}$$

is unbounded?

--J

 

[matt grime]

So, you've been shown the *proof* that it diverges - that the partial sums grow faster than summing a constant, yet you still think it ought to be convergent?

The problem is with the "logic" that adding up decreasing terms must converge. Nowhere in the definition of real numbers does it say that adding up decreasing numbers (tending to zero) must converge. That just isn't logic, it is a misapprehension you have about the real numbers.

A series converges iff the seqence of partial sums converges, that is the *definition*. By this example we can show your intuition doesn't fit the definition, so you need to reevaluate your intuition.