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Can \begin{equation} \sum_{k= 0}^{\infty}\frac{\left( -1\right) ^{k}}{k^2+a^{2}}, \ \ \sum_{k= 0}^{\infty}\frac{\left( -1\right) ^{k}}{(k-1/2)^2+a^{2}} \end{equation} be summed explicitly, where $a$ is a constant real number? Can any one give me some hint or tell me that the analytic expression doesn't exist. Thanks very much!

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$$\sum_{k=0}^\infty\frac{(-1)^k}{k^2+a^2}=\frac{1+a\pi\cdot\text{csch }(a\pi)}{2a^2}\qquad;\qquad\sum_{k=0}^\infty\frac{(-1)^k}{k^2-a^2}=-\frac{1+a\pi‌​\cdot\csc(a\pi)}{2a^2}$$ –  Lucian Mar 26 at 0:09
    
Maple can give closed forms for the sums. I have not tried Mathematics though. –  Mhenni Benghorbal Mar 26 at 22:49
    
@MhenniBenghorbal: they hopefully look like my answer. –  robjohn Mar 26 at 22:49
    
@robjohn: Maple gives the answers in terms of the digamma function as yours, however you can simplify further. –  Mhenni Benghorbal Mar 26 at 22:51
    
@MhenniBenghorbal: the answer to the second sum is in terms of the digamma function; the first sum is in terms of the hyperbolic cosecant function. –  robjohn Mar 26 at 22:53

3 Answers 3

up vote 3 down vote accepted

We can use equation $(7)$ from this answer: $$ \sum_{k\in\mathbb{Z}}\frac{1}{k+z}=\pi\cot(\pi z)\tag{1} $$ Directly from $(1)$, we have $$ \sum_{k\in\mathbb{Z}}\frac2{2k+z}=\pi\cot(\pi z/2)\tag{2} $$ Subtracting $(1)$ from $(2)$ yields $$ \begin{align} \pi\csc(\pi z) &=\sum_{k=-\infty}^\infty\frac{(-1)^k}{k+z}\tag{3}\\ &=\frac1z-2z\sum_{k=1}^\infty\frac{(-1)^k}{k^2-z^2}\tag{4} \end{align} $$ Substituting $z\mapsto iz$ in $(4)$ gives $$ \pi\,\mathrm{csch}(\pi z) =\frac1z+2z\sum_{k=1}^\infty\frac{(-1)^k}{k^2+z^2}\tag{5} $$ This works nicely to give us the first sum: $$ \color{#C00000}{\sum_{k=0}^\infty\frac{(-1)^k}{k^2+z^2} =\frac{\pi z\,\mathrm{csch}(\pi z)+1}{2z^2}}\tag{6} $$


Substituting $z\mapsto z+\frac12$ in $(3)$ gives $$ \begin{align} \pi\sec(\pi z) &=\sum_{k=-\infty}^\infty\frac{(-1)^k}{k+\frac12+z}\tag{7}\\ &=\sum_{k=0}^\infty\frac{(-1)^k(2k+1)}{\left(k+\frac12\right)^2-z^2}\tag{8} \end{align} $$ Substituting $z\mapsto iz$ in $(8)$ gives $$ \pi\,\mathrm{sech}(\pi z) =\sum_{k=0}^\infty\frac{(-1)^k(2k+1)}{\left(k+\frac12\right)^2+z^2}\tag{9} $$ This fails to give us the second sum because we have to add $\frac1{k+\frac12+z}$ and $\frac1{k+\frac12-z}$ instead of subtracting them because of the alternating signs. So we have to resort to another approach.


The digamma function $$ \psi(x)=-\gamma+\sum_{k=0}^\infty\left(\frac1{k+1}-\frac1{k+x}\right)\tag{10} $$ is the derivative of the logarithm of the Gamma function. $$ \psi\left(-\frac14+\frac{iz}{2}\right)-\psi\left(-\frac14-\frac{iz}{2}\right) =4iz\sum_{k=0}^\infty\frac1{\left(2k-\frac12\right)^2+z^2}\tag{11} $$ and $$ \psi\left(\frac14+\frac{iz}{2}\right)-\psi\left(\frac14-\frac{iz}{2}\right) =4iz\sum_{k=0}^\infty\frac1{\left(2k+\frac12\right)^2+z^2}\tag{12} $$ Subtracting $(12)$ from $(11)$ yields the second sum: $$ \begin{align} &\hspace{-1cm}\color{#C00000}{\sum_{k=0}^\infty\frac{(-1)^k}{\left(k-\frac12\right)^2+z^2}}\\[6pt] &\hspace{-1cm}\color{#C00000}{=\frac1{4iz}\left(\psi\left(-\frac14+\frac{iz}{2}\right)-\psi\left(-\frac14-\frac{iz}{2}\right)-\psi\left(\frac14+\frac{iz}{2}\right)+\psi\left(\frac14-\frac{iz}{2}\right)\right)}\tag{13} \end{align} $$


The second sum without alternation can be achieved by substituting $z\mapsto z-\frac12$ in $(1)$: $$ \begin{align} -\pi\tan(\pi z) &=\sum_{k\in\mathbb{Z}}\frac1{k-\frac12+z}\tag{14}\\ &=-2z\sum_{k=1}^\infty\frac1{\left(k-\frac12\right)^2-z^2}\tag{15} \end{align} $$ Substituting $z\mapsto iz$ in $(15)$ yields $$ \pi\tanh(\pi z)=2z\sum_{k=1}^\infty\frac1{\left(k-\frac12\right)^2+z^2}\tag{16} $$ Thus, we get the second sum without alternation $$ \sum_{k=0}^\infty\frac1{\left(k-\frac12\right)^2+z^2} =\frac4{4z^2+1}+\frac{\pi\tanh(\pi z)}{2z}\tag{17} $$ From $(12)$, we get $$ \sum_{k=0}^\infty\frac2{\left(2k+\frac12\right)^2+z^2} =\frac1{2iz}\left(\psi\left(\frac14+\frac{iz}{2}\right)-\psi\left(\frac14-\frac{iz}{2}\right)\right)\tag{18} $$ Subtracting $(18)$ from $(17)$ yields an alternate form for the second sum: $$ \begin{align} &\color{#C00000}{\sum_{k=0}^\infty\frac{(-1)^k}{\left(k-\frac12\right)^2+z^2}}\\[6pt] &\color{#C00000}{=\frac4{4z^2+1}+\frac{\pi\tanh(\pi z)}{2z}-\frac1{2iz}\left(\psi\left(\frac14+\frac{iz}{2}\right)-\psi\left(\frac14-\frac{iz}{2}\right)\right)}\tag{19} \end{align} $$

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It is well known that $$\csc z = \frac{1}{z} + 2z\sum_{k = 1}^\infty \frac{(-1)^k}{z^2 - k^2 \pi^2}.$$ Changing $z$ into $\pi z$ produces $$\pi\csc(\pi z) = \frac{1}{z} + 2z\sum_{k = 1}^\infty \frac{(-1)^k}{z^2 - k^2}, \quad z \ne n \pi, \quad n \in \mathbb{Z},$$ so $$\frac{\pi\csc(\pi z) - 1/z}{2z} = \sum_{k = 1}^\infty \frac{(-1)^k}{z^2 - k^2}.$$ Then it can be shown, by replacing $z$ with $iz$ and using $\csc(iz) = -i\operatorname{cosech} z$, that $$\operatorname{cosech} z = \frac{1}{z} + 2z\sum_{k = 1}^\infty \frac{(-1)^k}{z^2 + k^2}, \quad z \ne n \pi i, \quad n \in \mathbb{Z},$$ or $$\frac{\pi\operatorname{cosech}(\pi z) - 1/z}{2z} = \sum_{k = 1}^\infty \frac{(-1)^k}{z^2 + k^2}.$$

The second series looks familiar, but I cannot remember what it equals.

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The $k=0$ term is missing, but otherwise, this gives the correct answer for the first sum. –  robjohn Mar 27 at 0:15

$$\sum_{k= 0}^{\infty}\frac{\left( -1\right) ^{k}}{k^2+a^{2}}$$ This series converges to $$\dfrac{1}{2a^2}+\dfrac{\pi\mathrm{csch}(\pi a)}{2a}$$ Now, $$\sum_{k= 0}^{\infty}\frac{\left( -1\right) ^{k}}{(k-1/2)^2+a^{2}}$$ This series also converges by the comparison test.

Now, \begin{equation} \sum_{k= 0}^{\infty}\left(\frac{\left( -1\right) ^{k}}{k^2+a^{2}}+ \frac{\left( -1\right) ^{k}}{(k-1/2)^2+a^{2}}\right) \end{equation} Use the comparison test to see that this is convergent.

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I think the asker wanted to find explicitely the sum of each series (individually), rather than just show that they are convergent. I could have misunderstood though –  Hayden Mar 26 at 0:01
    
@Hayden Maybe. I will edit my answer. –  Sanath Mar 26 at 0:02
    
@Hayden See my edited answer. –  Sanath Mar 26 at 0:07
    
Are you sure about the divergence of the sum? Wolfram alpha says it converges by the comparison test. –  David H Mar 26 at 0:42
    
@DavidH My mistake - I wrote it in a hurry. –  Sanath Mar 26 at 0:48

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