How to evaluate the sum : $\sum_{k=1}^{n} \frac{k}{k^4+1/4}$

Question

I have been trying to figure out how to evaluate the following sum: $$S_n=\sum_{k=1}^{n} \frac{k}{k^4+1/4}$$

In the problem, the value of $S_{10}$ was given as $\frac{220}{221}$.

I have tried partial decomposition, no where I go. Series only seems like it telescopes, otherwise there isn't another way.

Any ideas are appreciated!

Answer 1

By Sophie Germain's identity $$ 4k^4+1 = (2k^2+2k+1)(2k^2-2k+1) \tag{A}$$ hence $$ \frac{1}{2k^2-2k+1}-\frac{1}{2k^2+2k+1} = \frac{4k}{4k^4+1} = \frac{k}{k^4+1/4}\tag{B} $$ and we may notice that by setting $p(x)=2x^2-2x+1$ we have $p(x+1)=2x^2+2x+1$.
In particular $$ \sum_{k=1}^{n}\frac{k}{k^4+1/4}=\sum_{k=1}^{n}\left(\frac{1}{p(k)}-\frac{1}{p(k+1)}\right) = \frac{1}{p(1)}-\frac{1}{p(n+1)}=1-\frac{1}{2n^2+2n+1} $$ equals $\frac{2n^2+2n}{2n^2+2n+1}$ for any $n\geq 1$.

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Telescoping is not strictly necessary to be able to compute the value of similar series. For instance $$ \sum_{k\geq 0}\frac{1}{k^4+4} = \frac{\pi\cos\pi+\sinh\pi}{8\sinh\pi}, $$ but this is a different story, related with Weierstrass products, the Poisson summation formula or the (inverse) Laplace transform.

Answer 2

Try to break the denominator into product of two factors:

$$\begin{align} 4k^4 + 1 &= (2k^2)^2 + 1 + 2 (2k^2) - 2 (2k^2) \\ &= (2k^2 +1)^2 - (2k)^2 \\ &= (2k^2 +2k +1)(2k^2 -2k+1) \end{align}$$

Using this we see the general term as:

$$T_k = \dfrac{1}{2k^2-2k+1} - \dfrac{1}{2k^2+2k+1} \\ T_{k+1} = \dfrac{1}{2k^2+2k+1} - \dfrac{1}{2(k+1)^2+2(k+1)+1} $$

Alternate terms cancel and the sum telescopes to:

$$1-\frac{1}{2n^2+2n+1}$$

Answer 3

The solutions by Jack d'Aurizio and samjoe are no doubt optimal, but I wondered if one could discover a solution even if one were not so clever as to see the (Sophie Germain) factorization. Here are the values of $S_n = \sum_{k=1}^{n}\frac{k}{k^4+1/4}$ for $n \le 15$: $$\frac{4}{5},\frac{12}{13},\frac{24}{25},\frac{40}{41},\frac{60}{61},\frac{84}{85}, \frac{112}{113},\frac{144}{145},\frac{180}{181},\frac{220}{221},\frac{264}{265},\frac{ 312}{313},\frac{364}{365},\frac{420}{421},\frac{480}{481}. $$ One notices that the numerators are divisible by 4; dividing by 4, one recognizes the sequence of triangular numbers, $t_n = n(n+1)/2$. Thus one can guess that $S_n = \frac{4 t_n}{4 t_n + 1}$. Then one can set out to prove this guess by induction. The induction step requires proving the identity $$\frac{4 t_n}{4 t_n + 1} + \frac{4(n+1)}{4(n+1)^4 + 1} = \frac{4 t_{n+1}}{4 t_{n+1} + 1},$$ or equivalently $$\frac{4(n+1)}{4(n+1)^4 + 1} = \frac{4 t_{n+1}}{4 t_{n+1} + 1} -\frac{4 t_n}{4 t_n + 1}.$$ It is straightforward to verify this by simplifying the RHS, and in the course of this, one discovers the Sophie Germain factorization.