Find $ \lim\limits_{{n \to \infty}} \frac1{2^n} \sum\limits_{k=1}^n \frac1{\sqrt{k}} \binom nk$

Question

Find $$\lim_{n \to \infty} \frac {1}{2^n} \sum_{k=1}^n \frac{1}{\sqrt{k}} \binom{n}{k}.$$

First time I thought about Stirling's approximation but didn't get anything by applying it. I would also think about a Riemann Sum, but no idea how to rewrite...

The answer is $0$.

Answer 1

Let $a_k = k^{-1/2}$. Notice that $(a_k)$ decreases to $0$. Then for each fixed $m \geq 1$ and for all $n \geq m$,

$$ \frac{1}{2^n} \sum_{k=1}^{n} \binom{n}{k} a_k \leq \frac{1}{2^n} \underbrace{\sum_{k=1}^{m} \binom{n}{k} (a_k - a_m)}_{= \mathcal{O}(n^m)} + \frac{1}{2^n} \underbrace{\sum_{k=1}^{n} \binom{n}{k} a_m}_{=(2^n - 1)a_m}. $$

Taking limsup as $n\to\infty$, it follows that

$$ \limsup_{n\to\infty} \frac{1}{2^n} \sum_{k=1}^{n} \binom{n}{k} a_k \leq a_m $$

Since $a_m \to 0$ as $m\to\infty$, this proves

$$ \lim_{n\to\infty} \frac{1}{2^n} \sum_{k=1}^{n} \binom{n}{k} a_k = 0. $$

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Addendum. I just saw that OP is a high-school student. Here is a little tweak of the argument above that does not use any fancy analysis stuffs.

Let $m_n = \lfloor \log n \rfloor$. Then for $n \geq 3$, we always have $1 \leq m_n \leq n$. Then

\begin{align*} 0 \leq \frac{1}{2^n} \sum_{k=1}^{n} \binom{n}{k} \frac{1}{\sqrt{k}} &= \frac{1}{2^n} \sum_{k=1}^{m_n} \binom{n}{k} \frac{1}{\sqrt{k}} + \frac{1}{2^n} \sum_{k=m_n + 1}^{n} \binom{n}{k} \frac{1}{\sqrt{k}} \\ &\leq \frac{1}{2^n} \sum_{k=1}^{m_n} n^k + \frac{1}{2^n} \sum_{k=m_n + 1}^{n} \binom{n}{k} \frac{1}{\sqrt{m_n}} \tag{1} \\ &\leq \frac{n^{1+m_n}}{2^n} + \frac{1}{\sqrt{m_n}}. \tag{2} \end{align*}

For $\text{(1)}$ I utilized the fact that $\binom{n}{k} \leq n^k$ and $\frac{1}{\sqrt{k}}$ is decreasing. For $\text{(2)}$ I utilized the geometric sum formula and the identity $\sum_{k=0}^{n} \binom{n}{k} = 2^n$.

Now taking $n\to\infty$ and applying the squeezing lemma proves the claim.

Answer 2

It is enough to apply Laplace's method.

Through the inverse Laplace transform we have $\frac{1}{\sqrt{k}}=\int_{0}^{+\infty}\frac{e^{-ks}}{\sqrt{\pi s}}\,ds $, hence

$$ \sum_{k=1}^{n}\binom{n}{k}\frac{1}{\sqrt{k}} = \int_{0}^{+\infty}\frac{-1+(1+e^{-s})^n}{\sqrt{\pi s}}\,ds =\frac{2}{\sqrt{\pi}}\int_{0}^{+\infty}\left[(1+e^{-s^2})^n-1\right]\,ds$$ where the last integrand function behaves like $(2^n-1) e^{-ns^2/2}$.
It follows that $$ \sum_{k=1}^{n}\binom{n}{k}\frac{1}{\sqrt{k}} \approx (2^n-1)\sqrt{\frac{2}{n}} $$ and the wanted limit is simply zero.

Answer 3

Using Abel's summation and the fact that $\sum \limits_{k=1}^{n} \binom{n}{k} = 2^{n}-1$ you can solve the above question.

From the Abel's summation we can say that $\sum \limits_{k=1}^{n} \frac{\binom{n}{k}}{\sqrt{k}} \approx \frac{\sum \limits_{k=1}^{n} \binom{n}{k}}{\sqrt{n}}-\int \limits_{1}^{n} ( \sum \limits_{k=1}^{t} \binom{t}{k} \frac{d}{dx}(\sqrt{t})) dt $

By substituting we arrive at : $ \frac{2^{n}-1}{\sqrt{n}}-\int \limits_{1}^{n} ( 2^{t}-1) \frac{d}{dx}(\sqrt{t}) dt $

It easy by comparison tests to bound the integral part by $\frac{2^n}{\sqrt{n}}$.

So its at most between $0$ and $2\frac{2^n}{\sqrt{n}}$ which both approach $0$ when divided by $2^n$ when $n \to \infty$.

Answer 4

$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \lim_{n \to \infty}\bracks{{1 \over 2^{n}}\sum_{k = 1}^{n} {{n \choose k} \over \root{k}}} & = \lim_{n \to \infty}\bracks{{1 \over 2^{n}}\sum_{k = 1}^{n} {n \choose k}\,{1 \over \root{\pi}}\int_{0}^{\infty}t^{-1/2}\expo{-kt}\,\dd t} \\[5mm] & = {1 \over \root{\pi}}\lim_{n \to \infty}\bracks{{1 \over 2^{n}}\int_{0}^{\infty}{\pars{1 + \expo{-t}}^{n} - 1 \over t^{1/2}}\,\dd t} \\[5mm] & = {1 \over \root{\pi}}\lim_{n \to \infty}\braces{{1 \over 2^{n - 1}} \int_{0}^{\infty}\bracks{{\pars{1 + \expo{-t^{2}}}^{n} - 1}}\,\dd t} \label{1}\tag{1} \\[5mm] & = {1 \over \root{\pi}}\lim_{n \to \infty}\bracks{{1 \over 2^{n - 1}} \int_{0}^{\infty}\pars{2^{n} - 1} \exp\pars{-\,{2^{n - 1} \over 2^{n} - 1}\,n\,t^{2}}\,\dd t} \label{2}\tag{2} \\[5mm] & = \bbx{\ds{0}} \end{align}

In line \eqref{1}, I used the Laplace Method to 'arrive' to line \eqref{2}.

Answer 5

The limit is part of a broad class of limits for which the classical analysis designed Toeplitz theorem. The theorem says that given an array of real numbers, $\lambda_{n,k}: 1\le k\le n,\ n\ge1 $, such that: $i) \ \lim_{n\to\infty}\lambda_{n,k}=0, \ \forall k\in \mathbb{N}$, $ii) \lim_{n\to\infty} \sum_{k=1}^n \lambda_{n,k}=1$ and $iii)$ there exists $C>0$ such that for all positive integers $n$, $\sum_{k=1}^n |\lambda_{n,k}|\le C$, then for any convergent sequence $\delta_k$, we have

$$\lim_{n\to\infty} \sum_{k=1}^n \lambda_{n,k} \delta_k=\lim_{n\to\infty}\delta_n .$$

In your problem set $\displaystyle \lambda_{n,k}=\frac{1}{2^n}\binom{n}{k}$ and $\displaystyle \delta_k=\frac{1}{\sqrt{k}}$, and since the conditions are satisfied, we have

$$\displaystyle \lim_{n \to \infty} \frac {1}{2^n} \displaystyle \sum_{k=1}^n \frac{\displaystyle \binom{n}{k}}{\sqrt{k}}=\lim_{n\to\infty} \frac{1}{\sqrt{n}}=0,$$ which ends the little proof.

Answer 6

$$\lim_{n \to \infty} \frac {1}{2^n} \sum_{k=1}^n \frac{{n}\choose{k}}{\sqrt{k}}=0$$

I offer a sketch proof. The basic idea is as follows: for large $n$, the small values of $k$ will make small contributions, because $\frac{{n \choose k}}{2^n}$ will be small there, and the large values of $k$ will make small contributions because $\sqrt{\frac{1}{k}}$ will be small there. Thus, the whole sum should be small too.

• Write $f(x)=x^{-1/2}\mathbb{I}(x\ge 1)$, and let $X$ denote a binomial random variable with parameter $1/2$ and $n$ trials.
• Then, the term we are taking limits of is $\mathbb{E}f(X)$
• The Central Limit Theorem says that, for large $n$, we have that $X\approx N(n/2,n/4)$ as distributions.
• We can thus (handwaving a bit) deduce that $$\frac{{n}\choose{k}}{2^n} \approx\sqrt{\frac{2}{\pi n}}\exp\left(-\frac{(2k-n)^2}{2n}\right)$$
  • One could also establish this via Stirling's formula, or likely other methods, if need be.
• Take some $\epsilon\in(0,1/3)$, and split the sum up into $1\le k\le (1-\epsilon)\frac{n}{2}, k>(1-\epsilon)\frac{n}{2}$.
  • In the first case, we have $$(2k-n)^2 \geq \epsilon^2n^2 \implies \exp\left(-\frac{(2k-n)^2}{2n}\right) \le \exp\left(-\frac{n\epsilon^2}{2}\right).$$ Thus, approximately, $$\frac {1}{2^n} \sum_{k=1}^{(1-\epsilon)\frac{n}{2}} \frac{{n}\choose{k}}{\sqrt{k}} \le \sum_{k=1}^{(1-\epsilon)\frac{n}{2}}\sqrt{\frac{2}{\pi n k}}\exp\left(-\frac{n\epsilon^2}{2}\right) \le(1-\epsilon)\sqrt{\frac{n}{2\pi}}\exp\left(-\frac{n\epsilon^2}{2}\right)\to 0$$
  • For the remainder of the sum, we have $$\frac {1}{2^n} \sum_{k>(1-\epsilon)n/2} \frac{{n}\choose{k}}{\sqrt{k}} \le \sum_{k>(1-\epsilon)n/2} \frac{{n}\choose{k}}{2^n}\sqrt{\frac{2}{{(1-\epsilon)n}}} \le \sum_{k\ge0} \frac{{n}\choose{k}}{2^n}\sqrt{\frac{2}{{(1-\epsilon)n}}}=\sqrt{\frac{2}{{(1-\epsilon)n}}}\to 0.$$

Thus, as desired, the limit is $0$.