Sequence of positive real numbers $(a_k)$ s.t. $\lim_{k \to \infty} a_k = 1$ and $\lim_{n \to \infty} a_1 a_2 \dots a_n = 0$

Question

I am looking for an example of a sequence of positive real numbers $(a_k)$ with $\lim_{k \to \infty} a_k = 1$ such that the sequence $(p_n)$ defined as $p_n=a_1 a_2 \dots a_n$ has limit 0 as $n \to \infty$.

Can anyone provide me with a concrete example, or maybe some hint or useful property of such a sequence?

Answer 1

Let consider

$$a_k=\frac{k}{k+1}$$

then

• $a_k \to 1$
• $\prod a_i =\frac12\frac23...\frac{k-1}k\frac{k}{k+1}=\frac1{k+1}\to 0$

Answer 2

Equivalently you're looking for $b_k = \ln (a_k)$ such that $b_k\to 0$ and $\sum_k b_k\to -\infty$.

$b_k=-\frac 1k$ fits the bill, yielding $a_k = e^{-1/k}$.

Answer 3

Hint. Consider a sequence such that $0<a_k <1$ for all $k$, but still approaches $1$. This will ensure that the product becomes smaller and smaller.

However, you're trying to balance things. If $a_k$ converges too fast to $1$, then the product will not diverge to zero (It will be decreasing, but bounded above by some $\varepsilon > 0$).

Answer 4

Any $a_k$ such that $0<a_k<1$ and $\sum (1-a_k)$ diverges will do.