If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $

I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.

The question does not especify the sign of $a_n$ but proving it for positives would already help.

I think the sign can change as long as the partial sums never get to be zero. Like if the series has only negative numbers the result seems to stil hold.

Show that the series $\sum_{n=1}^\infty \frac{a_n}{S_n}$ diverges

Solution 1:

This problem has an interesting history from the early days of real analysis attached to it. I really like this story and the proof it contains of the problem in this question is also different from the other answers posted.

In 1827 a M. Oliver published a proof in Crelle's Journal (Journal für die reine und angewandte Mathematik) of the following theorem:

Theorem M. Oliver: A series of positive terms converges if and only if $\lim_{n\to\infty} na_n = 0$.

Not long after this paper had been published Niels Henrik Abel responded with a short note noting that the theorem "does not seem to hold true" as the series $\sum_{n=2}^\infty\frac{1}{n\log(n)}$ diverges but $\lim_{n\to\infty}n\cdot \frac{1}{n\log(n)} = 0$ contradicting the theorem above.

Abel then followed it up by another short note showing that there cannot exist any function $\phi(n)$ such that

$$\sum_{k=1}^\infty a_n ~~~\text{converges} \iff \lim_{n\to\infty}\phi(n)a_n = 0$$

The proof is remarkably simple and uses the lemma below (which coincides with the problem this question asks for). I will here present Abel's original proof. It's from memory so it's not word for word but the main idea behind the proof should be intact.

Lemma: if a series of positive terms $\sum_{k=1}^\infty a_k$ diverges then so does the series $\sum_{k=1}^\infty \frac{a_k}{a_1+a_2+\ldots + a_{k-1}}$.

Proof: Let $s_n = \sum_{k=1}^{n}a_k$, then

$$\log\left(\frac{s_k}{s_{k-1}}\right) = \log\left(1+\frac{a_k}{s_{k-1}}\right) < \frac{a_k}{s_{k-1}}$$

since $\log(1+x) < x$. Summing over $k=2,3,\ldots,n$ the left hand side telescopes to give

$$\log\left(\frac{s_n}{s_{1}}\right) < \sum_{k=1}^n \frac{a_k}{s_{k-1}}$$

Now if $s_n = \sum_{k=1}^n a_k$ diverges then so does the logarithm of $s_n$ and it follows that $\sum_{k=1}^n \frac{a_k}{s_{k-1}}$ also diverges.

Proof of Abel's theorem above: Assume there exist a function $\phi(n)$ with the properties that $\sum a_k$ converges if and only if $\lim_{n\to\infty}\phi(n)a_n = 0$. Then $\sum_{k=1}^\infty \frac{1}{\phi(k)}$ diverges since $\lim_{n\to\infty} \phi(n)\frac{1}{\phi(n)} = 1$. By the lemma above this implies that $$\sum_{k=1}^\infty \frac{\frac{1}{\phi(k)}}{\frac{1}{\phi(1)}+\frac{1}{\phi(2)} + \ldots + \frac{1}{\phi(k-1)}}$$ also diverges but $$\lim_{n\to\infty} \phi(n)\frac{\frac{1}{\phi(n)}}{\frac{1}{\phi(1)}+\ldots + \frac{1}{\phi(n-1)}} = \lim_{n\to\infty}\frac{1}{\frac{1}{\phi(1)}+\ldots + \frac{1}{\phi(n-1)}} = 0$$

which would indicate that it is convergent. No such function $\phi(n)$ can therefore exist.

Solution 2:

Suppose the $a_n$ are non-negative. To prove that the series diverges to $+\infty$, it is enough to prove that for every $N\in\Bbb N$, there exists $N'>N$ such that $$\sum_{n=N}^{N'}\frac{a_n}{a_0+\cdots+a_n}\geq \frac12$$ So fix $N>0$, and notice that for every $M>N$ $$\sum_{n=N}^{M}\frac{a_n}{a_0+\cdots+a_n}\geq\sum_{n=N}^{M}\frac{a_n}{a_0+\cdots+a_{M}}=\frac{a_N+\cdots+a_{M}}{a_0+\cdots+a_{M}}=1-\frac{C}{a_0+\cdots+a_{M}}$$ where $C=a_0+\cdots+a_{N-1}$ is a constant. The second term is of the form "constant over something that tends to infinity" and so tends to $0$, and eventually we get a minoration by $\frac12$, showing that the series diverges to infinity.

Solution 3:

Intuition comes from the continuous analogue: if $f : [a, \infty) \to \Bbb{R}^+$ is a positive piecewise continuous function such that $\int_{a}^{\infty} f(x) \, dx = \infty$, then

$$ \int_{a}^{\infty} \frac{f(x)}{F(x)} \, dx = \log F(\infty) - \log F(a) = \infty $$

where $F$ is an antiderivative of $f$ (which is also positie on $[a, \infty)$. Bringing this idea to discrete setting is not hard. Indeed, when $a_k$'s are positive then you may apply Stolz theorem to

$$ \lim_{n\to\infty} \frac{\sum_{k=1}^{n} (a_k / s_k)}{\log s_n}. \tag{1} $$

More precisely, we have either $\limsup_{n\to\infty} a_n / s_n > 0$ or that (1) converges to 1, and in either case the series diverges.