Why won't a series converge if the limit of the sequence is 0?

Question

Just thinking about it in terms of logic, shouldn't the series of a sequence whose limit as $n$ approaches infinity is 0 converge?

I know that the $n$th term test for divergence says that if a series is convergent, then the limit of its sequence is 0 and I also know there are some sequences for which it has been "proven" that their series does not converge even though the sequence converges to 0, but I just don't believe these tests. If we stretch $n$ out to infinity and the terms are approaching 0, then how is it possible for the sum of the terms to "overflow" and diverge if the terms are becoming negligibly small?

Answer 1

A very easy counterexample would be $$ 1, \underbrace{\frac12, \frac12}_{2\text{ halves}}, \underbrace{\frac13, \frac13, \frac13}_{3\text{ thirds}}, \underbrace{\frac14, \frac14, \frac14, \frac14}_{4\text{ fourths}}, \underbrace{\frac15, \frac15, \frac15, \frac15, \frac15}_{5\text{ fifths}}, \ldots $$ This sequence clearly converges to $0$, but if you try to sum it, it should be obvious that it has partial sums as large as you'd like them to be -- so the series diverges.

Try whichever argument you have in mind for believing that the series should converge, and attempt to figure out why it doesn't work for this one.

Answer 2

Do you think the series

$$1+\frac12+\frac12 + \frac14+\frac14+\frac14+\frac14 + \frac18 + \cdots$$

converges? Note, there are $2$ terms equal to $\frac12$, $4$ terms equal to $\frac14$, $8$ terms equal to $\frac18$ and so on, with $2^i$ terms equal to $\frac{1}{2^i}$ for each $i\in\mathbb N$.

You will probably agree that this series diverges. In fact, if you give me a number $m\in \mathbb N$, I can calculate exactly how many terms of the series you have to add together for the sum to reach $m$.

For example, it takes $1$ term to reach $1$, it takes $1+2=3$ terms to reach $2$, and then $1+2+4=7$ terms to reach $3$. You can show, with a simple inductive argument, that you will reach $m$ after

$$1+2+4+\dots + 2^{m-1}$$

terms, which is actually equal to $2^m-1$ and is certainly a finite number.

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It's good to understand the concept why this series diverges. The thing is that yes, the terms go to $0$, but they don't do so "fast enough". The problem is than once the terms hit $\frac14$, they stick at that number for $4$ steps, long enough for the sum to increase by $1$.

And imagine what happens way way way down the line. The sum is equal to $\frac{1}{1024}$ for a whole $1024$ terms, for example. Sure, it will eventually fall to an even lower number, but it will stay on that number for even longer, again long enough for the whole sum to increase by $1$.

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Side fact: the series I wrote down at the start has the bonus property that each term in the sequence is larger than the corresponding term of the sequence

$$1+\frac12+\frac13+\frac14+\frac15+\cdots$$

which is also known as the harmonic series and is the most famous divergent series. So, you now see that if you sum $2^m$ terms of the harmonic series, your sum will be equal to at least $m$ (and more, in fact).

Answer 3

However note that in the field of $p$-adic numbers $\mathbf Q_p$, a series $\sum_na_n$ converges if and only if the sequence $(a_n)$ tends to $0$.

Answer 4

I will assume that Fundamental theorem of calculus is believable enough.

While I could relate to you that ratio or root tests are not that easy to understand intuitively, it doesn't get more visual than integral test for convergence. I mean, look at this. You can immediately see that $$\int_1^{n+1}\frac{dx}x\leq\sum_{k=1}^n\frac 1 k$$ Employ Fundamental theorem to get $$\ln(n+1)\leq\sum_{k=1}^n\frac 1 k$$ so, harmonic series diverges because logarithm is unbounded. And why is logarithm unbounded? Because it is the inverse of $\exp\colon \mathbb R\to \mathbb R_+$.

Answer 5

You are echoing a 2-millenia-old quesite: will Achilles overtake the Tortoise ? , and
if Achilles decelerates,... Convergence/divergence of infinite sequences and series took long time to be mastered somehow, also because it was clear that the result of adding many small terms was depending on how many vs. how small.
Just consider Stereographic Projection and the philosophical debates it arouses.

Answer 6

Just look at the problem from the other end.

Take a function like $f(x) = \sqrt{x}$ or $\log x$ - they grow infinitely with the $x$ growth, though their growth ratio decreases (their first derivatives tend to zero).

Consider the function increments for equal steps of argument, say, for arguments natural: $$\Delta f = f(n+1) - f(n)\quad \text{ for } n\in\mathbb N^+$$ As the $f$ function grows slower and slower, these deltas get smaller and smaller for growing $n$, and they actually tend to zero: for arbitrarily small $\varepsilon > 0$ there exists $n$ big enough, for which $\Delta f < \varepsilon$.

Now $\Delta f$ defines a sequence convergent to zero, whose partial sums are values of $\sqrt n$, that is a function growing with no bound, hence a divergent series.