# Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$

Sis.

• Boy, I went to the wrong high school! Feb 18 '13 at 14:48
• My next question on the Riemann Hypothesis will be called "elementary school question". Feb 18 '13 at 14:53
• Where is this high school? Feb 18 '13 at 14:58
• @Chris'ssisterandpals: I see. Feb 18 '13 at 15:02
• I feel stupid now. I only got to this stuff at university level. Feb 19 '13 at 3:25

Here is a high school level answer: \begin{align} \sum_{k=1}^n\left(\frac1{(2k-1)^2}-\frac3{4k^2}\right) &=\sum_{k=1}^n\left(\frac1{(2k-1)^2}+\frac1{(2k)^2}-\frac1{k^2}\right)\\ &=\sum_{k=1}^{2n}\frac1{k^2}-\sum_{k=1}^n\frac1{k^2}\\ &=\sum_{k=n+1}^{2n}\frac1{k^2}\tag{1} \end{align} Using partial fractions and summing the telescoping series, we get $$\hspace{-1cm} \frac1{n+1}-\frac1{2n+2} =\sum_{k=n+1}^{2n}\frac1{k(k+1)} \le\sum_{k=n+1}^{2n}\frac1{k^2} \le\sum_{k=n+1}^{2n}\frac1{k(k-1)} =\frac1n-\frac1{2n}\tag{2}$$ Therefore, the Squeeze Theorem and $(2)$ yield $$\lim_{n\to\infty}n\sum_{k=n+1}^{2n}\frac1{k^2}=\frac12\tag{3}$$

• Oh, this is a candidate for the penultimate high school year since you used no integral. Just awesome! (+1) Actually, excepting the limit part, it could also be a good problem for the middle school. Feb 18 '13 at 17:08
• @Chris'ssisterandpals: I don't usually post two answers to the same question, but the prerequisite level of these answers was so disparate, that I felt I should.
– robjohn
Feb 18 '13 at 17:20
• I have no problem with that. For me the answers are important! :-) Feb 18 '13 at 17:22
• Very nice!${}{}$
– mrf
Feb 19 '13 at 11:56
• (+1) The easier, the better. Sep 13 '14 at 18:01

OK, it turns out that

$$\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right) = \sum_{k=1}^{n-1} \frac{1}{(k+n)^2}$$

This may be shown by observing that

$$\sum_{k=1}^n \frac{1}{(2k-1)^2} = \sum_{k=1}^{2 n-1} \frac{1}{k^2} - \frac{1}{2^2} \sum_{k=1}^n \frac{1}{k^2}$$

The desired limit may then be rewritten as

$$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n-1} \frac{1}{(1 + (k/n))^2}$$

which you may recognize as a Riemann sum, equal to

$$\int_0^1 dx \: \frac{1}{(1+x)^2} = \frac{1}{2}$$

• nice work and well shown. (+1) Feb 18 '13 at 15:18
• This is the way I wanted to go, but I went with the Euler-Maclaurin Sum Formula as an alternative :-) (+1)
– robjohn
Feb 18 '13 at 16:46
• Nice to see it done different ways. Feb 18 '13 at 16:56
• I have added yet another way :-)
– robjohn
Feb 18 '13 at 17:15
• You're an animal, @robjohn! Feb 18 '13 at 20:09

$$n\sum_{k=1}^n\frac{1}{(2k-1)^2}-\frac{3}{4k^2} =n(H_{2n}^{(2)}-H_{n}^{(2)})=\sum_{j=1}^n\frac{n}{(j+n)^2}\to\int_0^1\frac{dx}{(1+x)^2} =\frac{1}{2}$$

• This is incorrect. The limit indeed converges, but not to the limit you stated. Feb 18 '13 at 14:58
• Could you tell me where I made a mistake? Feb 18 '13 at 15:01
• I think he is right. Feb 18 '13 at 15:13
• @Chis's sister and pals Where do you get these questions? Feb 18 '13 at 15:20
• @IshanBanerjee: I have them from my brother. Feb 18 '13 at 16:12

Using the Euler-Maclaurin Sum Formula, \begin{align} \sum_{k=1}^n\left(\frac1{(2k-1)^2}-\frac3{4k^2}\right) &=\sum_{k=1}^n\left(\frac1{(2k-1)^2}+\frac1{(2k)^2}-\frac1{k^2}\right)\\ &=\sum_{k=1}^{2n}\frac1{k^2}-\sum_{k=1}^n\frac1{k^2}\\ &=\left(C-\frac1{2n}+O\left(\frac1{n^2}\right)\right)-\left(C-\frac1n+O\left(\frac1{n^2}\right)\right)\\ &=\frac1{2n}+O\left(\frac1{n^2}\right) \end{align} Therefore, \begin{align} \lim_{n\to\infty}n\sum_{k=1}^n\left(\frac1{(2k-1)^2}-\frac3{4k^2}\right) &=\lim_{n\to\infty}n\left(\frac1{2n}+O\left(\frac1{n^2}\right)\right)\\ &=\frac12 \end{align}

• Another good shot with Euler-Maclaurin Sum Formula. Thanks John! (+1) Feb 18 '13 at 16:51
• @Chris'ssisterandpals: I am writing up a high school level answer now.
– robjohn
Feb 18 '13 at 16:52

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\lim_{n\ \to\ \infty}\bracks{n\sum_{k = 1}^{n}\fermi\pars{k}}:\ {\large ?}. \qquad\fermi\pars{k} \equiv {1 \over \pars{2k -1}^{2}} - {3 \over 4k^{2}}}$

Note that $\ds{\sum_{k = 1}^{\infty}\fermi\pars{k} = 0}$ because $$\sum_{k = 1}^{\infty}\fermi\pars{k} =\sum_{k = 1}^{\infty}{1 \over \pars{2k - 1}^{2}} -\sum_{k = 1}^{\infty}{3 \over 4k^{2}} =\bracks{\sum_{k = 1}^{\infty}{1 \over k^{2}} -\sum_{k = 1}^{\infty}{1 \over \pars{2k}^{2}}} -\sum_{k = 1}^{\infty}{3 \over 4k^{2}}$$

With Stolz-Cesàro Theorem: \begin{align}&\color{#66f}{\large\lim_{n\ \to\ \infty}\bracks{% n\sum_{k = 1}^{n} \fermi\pars{k}}}=\lim_{n\ \to\ \infty}{\sum_{k = 1}^{n}% \fermi\pars{k} \over 1/n} =\lim_{n\ \to\ \infty}{% \sum_{k = 1}^{n + 1}\fermi\pars{k} - \sum_{k = 1}^{n}\fermi\pars{k} \over 1/\pars{n + 1} - 1/n} \\[3mm]&=\lim_{n\ \to\ \infty}\bracks{-n\pars{n + 1}\fermi\pars{n + 1}} =\lim_{n\ \to\ \infty}\braces{n\pars{n + 1} \bracks{{3 \over 4\pars{n + 1}^{2}} - {1 \over 4\pars{n + 1}^{2}}}} \\[3mm]&=\half\,\lim_{n\ \to\ \infty}{n \over n + 1}=\color{#66f}{\Large\half} \end{align}

• Good job there! (+1) Sep 13 '14 at 6:30
• @Chris'ssis Thanks. You are always posting interesting questions. Sep 13 '14 at 6:40