What is $\lim_{n \to \infty} (\sum\limits_{j=0}^n {2n \choose j}(1+p)^j)^\frac{1}{2n}$

Question

I'm trying to find the limit:

$$L= \lim_{n \to \infty} \left(\sum\limits_{j=0}^n {2n \choose j}(1+p)^j\right)^\frac{1}{2n}$$

We know that:

$$\left( \sum\limits_{j=0}^{2n} {2n \choose j}(1+p)^j\right)^\frac{1}{2n}=2+p$$

I suspect:

$$\lim_{n \to \infty}\left( \sum\limits_{j=n+1}^{2n} {2n \choose j}(1+p)^j\right)^\frac{1}{2n}=2+p$$ as well. So, I suspect $L<2+p$. Not sure how to proceed with it. Any numerical trick to find it for really large values of $n$?

Answer 1

I will assume that $p \geq 0$. Then noting the inequality

$$ \binom{2n}{n}(1+p)^n \leq \sum_{j=0}^{n} \binom{2n}{j}(1+p)^j \leq n \cdot \binom{2n}{n}(1+p)^n $$

and the limit $\lim_{n\to\infty} n^{1/n} = 1$, we deduce that

$$ L = \lim_{n\to\infty} \left[ \binom{2n}{n} (1+p)^n \right]^{\frac{1}{2n}} = 2\sqrt{1+p}. $$

AM-GM inequality guarantees that $L \leq 2+p$ and the equality holds if and only if $p = 0$.

Answer 2

Your guesses seem to be very correct but I do not have a proof for it.

What happens is that $$\sum_{j=0}^n \binom{2 n}{j} (1+p)^j=(2+p)^{2 n}-\binom{2 n}{n+1} (1+p)^{n+1} \, _2F_1(1,1-n;n+2;-(1+p))$$ where appears the gaussian hypergeometric function.

So, your expression is $$(2+p) \left(1-\binom{2 n}{n+1} \frac{(1+p)^{n+1}} { (2+p)^{2 n} }\, _2F_1(1,1-n;n+2;-(1+p))\right){}^{\frac{1}{2 n}}$$ that is to say $$(2+p) \left(1-\binom{2 n}{n+1} (1+p)^{n+1} \, _2F_1(n+1,2 n+1;n+2;-(1+p))\right){}^{\frac{1}{2 n}}$$

Numerically, the multiplying factor is an increasing function going to $1$ from below.

For illustration purposes, let $$a_n(p)=\left(1-\binom{2 n}{n+1} (1+p)^{n+1} \, _2F_1(n+1,2 n+1;n+2;-(1+p))\right){}^{\frac{1}{2 n}}$$ Computing for $p=1,2,\cdots,6$, we have the following values

$$\left( \begin{array}{ccccccc} n & a_n(1)& a_n(2)& a_n(3)& a_n(4)& a_n(5)& a_n(6)\\ 100 & 0.932448 & 0.855352 & 0.789691 & 0.735520 & 0.690481 & 0.652489 \\ 200 & 0.936834 & 0.859938 & 0.794147 & 0.739785 & 0.694553 & 0.656382 \\ 300 & 0.938512 & 0.861674 & 0.795826 & 0.741388 & 0.696081 & 0.657841 \\ 400 & 0.939418 & 0.862606 & 0.796725 & 0.742245 & 0.696898 & 0.658620 \\ 500 & 0.939992 & 0.863193 & 0.797290 & 0.742783 & 0.697410 & 0.659110 \\ 600 & 0.940390 & 0.863599 & 0.797681 & 0.743155 & 0.697764 & 0.659447 \\ 700 & 0.940684 & 0.863898 & 0.797968 & 0.743428 & 0.698024 & 0.659695 \\ 800 & 0.940910 & 0.864127 & 0.798188 & 0.743638 & 0.698223 & 0.659885 \\ 900 & 0.941091 & 0.864310 & 0.798363 & 0.743804 & 0.698382 & 0.660036 \\ 1000 & 0.941238 & 0.864458 & 0.798506 & 0.743940 & 0.698510 & 0.660158 \\ 1100 & 0.941360 & 0.864582 & 0.798624 & 0.744052 & 0.698617 & 0.660260 \\ 1200 & 0.941464 & 0.864687 & 0.798724 & 0.744147 & 0.698708 & 0.660346 \\ 1300 & 0.941553 & 0.864776 & 0.798810 & 0.744229 & 0.698785 & 0.660420 \\ 1400 & 0.941630 & 0.864854 & 0.798884 & 0.744299 & 0.698852 & 0.660484 \\ 1500 & 0.941698 & 0.864922 & 0.798950 & 0.744361 & 0.698911 & 0.660540 \\ 1600 & 0.941758 & 0.864982 & 0.799007 & 0.744416 & 0.698963 & 0.660589 \\ 1700 & 0.941811 & 0.865036 & 0.799058 & 0.744465 & 0.699009 & 0.660633 \\ 1800 & 0.941859 & 0.865084 & 0.799104 & 0.744508 & 0.699050 & 0.660673 \\ 1900 & 0.941902 & 0.865127 & 0.799146 & 0.744547 & 0.699088 & 0.660708 \\ 2000 & 0.941942 & 0.865167 & 0.799183 & 0.744583 & 0.699121 & 0.660741 \end{array} \right)$$