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$\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+1}+\dfrac{n}{n^2+2}+\dfrac{n}{n^2+3}\cdots\cdots+\dfrac{n}{n^2+n}\right)$

Can we write it as following

$E=\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+1}\right)+\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+2}\right)+\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+3}\right)\cdots\cdots+\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+n}\right)\tag{1}$

Let's see what happens:-

$$\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+1}\right)$$ $$\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n^2}}=0$$

In the same way for further terms, we will get $0$

Let's also confirm for general term

$$\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+n}\right)$$ $$\lim\limits_{n\to\infty}\left(\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n}}\right)=0$$

So the whole expression $E$ will be zero

But actual answer is $1$

Let's see what happens if we evaluate the original expression $OE=\lim\limits_{n\to\infty}\left(\dfrac{n}{n^2+1}+\dfrac{n}{n^2+2}+\dfrac{n}{n^2+3}\cdots\cdots+\dfrac{n}{n^2+n}\right)$

$OE=\lim\limits_{n\to\infty}\left(\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n^2}}+\dfrac{\dfrac{1}{n}}{1+\dfrac{2}{n^2}}+\dfrac{\dfrac{1}{n}}{1+\dfrac{3}{n^2}}\cdots\cdots+\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n}}\right)$

Now we can easily see that each term inside the bracket is tending to $0$, so can we say that sum of all terms upto infinity as well tends to zero?

I think we cannot because the quantity is not exactly zero, it is tending to zero, so when we add the values tending to zero upto infinity, we may not get zero.

But I got the following counter thought:-

$\lim\limits_{x\to0}\dfrac{(1+x)^\frac{1}{3}-1}{x}$

As we know $(1+x)^n=1+nx+\dfrac{n(n-1)}{2}x^2+\dfrac{n(n-1)(n-2)}{6}x^3\cdots\cdots\infty$ where $|x|<1$

$\lim\limits_{x\to0}\dfrac{\left(1+\dfrac{1}{3}x-\dfrac{1}{3}\cdot\dfrac{2}{3}\cdot\dfrac{1}{2}x^2+\dfrac{1}{3}\cdot\dfrac{2}{3}\cdot\dfrac{5}{3}\cdot\dfrac{1}{6}x^3\cdots\cdots\right)-1}{x}$

$\lim\limits_{x\to0}\dfrac{1}{3}-\dfrac{1}{3}\cdot\dfrac{2}{3}\cdot\dfrac{1}{2}x+\dfrac{1}{3}\cdot\dfrac{2}{3}\cdot\dfrac{5}{3}\cdot\dfrac{1}{6}x^2\cdots\cdots$

Now here also all the terms except $\dfrac{1}{3}$ are tending to $0$. So here also we can say that the whole quantity may not turn out to be zero as we are adding all terms upto infinity.

But surprisingly $\dfrac{1}{3}$ is the correct answer.

I am feeling very confused in these two things. Please help me.

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Observe that $$\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots+\frac{n}{n^2+n}$$ lies between $$\frac{n}{n^2+n}+\frac{n}{n^2+n}+\cdots+\frac{n}{n^2+n}=\frac{n}{n+1} $$ and $$\frac{n}{n^2}+\frac{n}{n^2}+\cdots+\frac{n}{n^2}=1.$$

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I will be giving a $\varepsilon -\mathcal{N}$ proof

notice

$$\begin{align}\frac{n}{1+n^2} + \frac{n}{2+n^2} + \cdots + \frac{n}{n+n^2} - 1= n\Big( \frac{1}{1+n^2} - \frac{1}{n^2} + \cdots + \frac{1}{n+n^2} - \frac{1}{n^2} \Big) = h_n\end{align}$$

Claim $\dfrac{n}{n+n^2} > \dfrac{r}{r+n^2}$


So

$|h_n| < \Big|n\cdot(\dfrac{1}{n^2}\cdot\dfrac{n^2}{n+n^2})\Big|=\dfrac{1}{n+1}<\dfrac{1}{n}$

so $\forall n > \dfrac{1}{\varepsilon}$ $|h_n| < \varepsilon$

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That is not legitimate, because the number of terms inside the limit are growing as $n$ tends to infinity.

Rather, one sees that \begin{align*} \sum_{k=1}^{n}\dfrac{n}{n^{2}+n}\leq\sum_{k=1}^{n}\dfrac{n}{n^{2}+k}\leq\sum_{k=1}^{n}\dfrac{n}{n^{2}+1}, \end{align*} the left and right-sided both tend to $1$.

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  • $\begingroup$ Ok then what about the second case $\endgroup$
    – prat
    Dec 8 '19 at 3:19
  • $\begingroup$ The number of terms do not grow as $x$ is running. $\endgroup$
    – user284331
    Dec 8 '19 at 3:20
  • $\begingroup$ but if you see the formula carefully, it goes till infinity $\endgroup$
    – prat
    Dec 8 '19 at 3:21
  • $\begingroup$ Yes, but the terms are still that much. If you see the first one, each times $n$ varies, there are $n$ terms going on, but the second case no, there is such an amount (assume you can really count them) no matter what $x$ is, both are in different circumstances. $\endgroup$
    – user284331
    Dec 8 '19 at 3:23
  • $\begingroup$ ok well said, but then what is the answer for this $\lim\limits_{x\to0}\dfrac{(1+x+x^2+x^3+x^4\cdots\cdots\infty)-1}{x}$? $\endgroup$
    – prat
    Dec 8 '19 at 3:26
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If you know harmonic numbers $$S_n=\sum_{i=i}^n \frac 1 {n^2+i}=H_{n^2+n}-H_{n^2}$$ Using the asympotics $$H_p=\gamma +\log \left({p}\right)+\frac{1}{2 p}-\frac{1}{12 p^2}+O\left(\frac{1}{p^3}\right)$$ apply it twice and continue with Taylor series to get $$n S_n=1-\frac{1}{2 n}-\frac{1}{6 n^2}+O\left(\frac{1}{n^3}\right)$$ which shows the limit nd how it is approached.

Moreover, this gives a quite good approximation of the sum. Using $n=10$, the exact value is $\frac{11210403701434961}{11818204429243212} \approx 0.94857$ while the above truncated series gives $\frac{569}{600}\approx 0.94833$.

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You can use the Squeeze Theorem, with

$$ n\cdot \frac{n}{n^2 + n} \leq a_n \leq n\cdot \frac{n}{n^2 + 1} $$

And as $ n \to \infty$, you get $ 1 \leq \lim_{n \to \infty} a_n \leq 1 $, So, using the Squeeze Theorem, $a_n \to 1$.

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  • $\begingroup$ But $\lim_{n\to\infty}an$ for n/n+1 sequence is 1(not 0). So will the series n/n+1 converge then? $\endgroup$ Dec 6 '20 at 14:41
  • $\begingroup$ I'm not sure what you mean, I've never wrote that the limit of something here is $0$. $\frac{n}{n+1}$ does converge to $1$. $\endgroup$
    – talbi
    Dec 6 '20 at 14:47
  • $\begingroup$ I mean that if a series has to converge, its nth term(when n is tending to infinity) has to tend to zero. But for your RHS, that isn't happening. $\endgroup$ Dec 6 '20 at 14:51
  • $\begingroup$ On the RHS, we have $n \cdot \frac{n}{n^2+1} = \frac{n^2}{n^2+1}$. This converges to $1$, not zero! Perhaps you missed the multiplication by $n$? $\endgroup$
    – talbi
    Dec 6 '20 at 14:54
  • $\begingroup$ My bad, LHS I meant! $\endgroup$ Dec 6 '20 at 15:07
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$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\on}[1]{\operatorname{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $\ds{\bbox[5px,#ffd]{}}$


\begin{align} &\bbox[5px,#ffd]{\sum_{k = 1}^{n}{n \over n^{2} + k}} = n\sum_{k = 0}^{n - 1}{1 \over k + 1 + n^{2}} \\[5mm] = &\ n\sum_{k = 0}^{\infty}\pars{{1 \over k + 1 + n^{2}} - {1 \over k + n + 1 + n^{2}}} = n\pars{H_{n^{2} + n} - H_{n^{2}}} \\[5mm] \stackrel{\mrm{as}\ n\ \to\ \infty}{\sim}\,\,\,& n\braces{\bracks{\ln\pars{n^{2} + n} + \gamma + {1 \over 2n^{2} + 2n}} - \bracks{\ln\pars{n^{2}} + \gamma + {1 \over 2n^{2}}}} \\[5mm] \stackrel{\mrm{as}\ n\ \to\ \infty}{\sim}\,\,\, &\ n\ln\pars{1 + {1 \over n}} \stackrel{\mrm{as}\ n\ \to\ \infty}{\to}\,\,\, \bbx{1} \\ & \end{align} $\ds{H_{z}}$ is a Harmonic Number and $\ds{\gamma}$ is the Euler-Mascheroni Constant.
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Following the logic in your answer $$ \lim_{n\to\infty}\overbrace{\left(\frac1n+\frac1n+\cdots+\frac1n\right)}^\text{$n$ terms}=0 $$ since each term tends to $0$. This is obviously false since the sum for each $n$ is $1$, so the limit is $1$.

We can say that a finite sum of terms which have a limit equals the finite sum of the limits. However, the same cannot be said for an unlimited sum of terms even though each has a limit. We must have some other theorem that allows us to swap order of the limits.

One proper way to handle the sum in the question is to bound the sum with $$ \frac{n}{n+1}\le\left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots+\frac{n}{n^2+n}\right)\le1 $$ and then the Squeeze Theorem says the limit is $1$.

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  • $\begingroup$ I think something wrong is there in your first statement, it should be $OE=\lim\limits_{n\to\infty}\left(\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n^2}}+\dfrac{\dfrac{1}{n}}{1+\dfrac{2}{n^2}}+\dfrac{\dfrac{1}{n}}{1+\dfrac{3}{n^2}}\cdots\cdots+\dfrac{\dfrac{1}{n}}{1+\dfrac{1}{n}}\right)$ and here I don't see that it is obviously $1$? Please correct me if I am wrong. $\endgroup$
    – prat
    Dec 8 '19 at 4:53
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    $\begingroup$ @prat: No, the first statement is a simpler example that shows why the logic used in your solution fails. The second statement shows how to see that the limit of the sum in the question is $1$ by applying the Squeeze Theorem. $\endgroup$
    – robjohn
    Dec 8 '19 at 5:04
  • $\begingroup$ yeah you are right, I only have question, What happened in the second case mentioned in the question, why over there same logic doesn't apply? $\endgroup$
    – prat
    Dec 8 '19 at 5:10
  • $\begingroup$ The sum is $$\sum_{n=0}^\infty\frac{\frac43-1}1\frac{\frac43-2}2\cdots\frac{\frac43-n}nx^n$$ the terms of which are dominated by $|x|^n$, so we can use Dominated Convergence to say the series will converge for $|x|\lt1$. $\endgroup$
    – robjohn
    Dec 8 '19 at 7:06

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