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How to find this sum? $$\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$$

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up vote 6 down vote accepted

As I'm a fan of proofs without words, here is my pictorial effort of why

$$\sum_{k=1}^{2n} (-1)^k k^2 = \sum_{i=1}^{2n} i = n(2n+1).$$

enter image description here

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Hint: What is $(2i)^2-(2i-1)^2$? Next, what are $\displaystyle\sum_{i=1}^n1$ and $\displaystyle\sum_{i=1}^ni$?

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Try working with the sequence $S_n=\sum_1^{2n}(-1)^kk^2$. If you look at the sequence of differences (sidrat hefreshim in hebrew, i'm not sure if that the correct translation. any hebrew speaker- please correct me here) of $A_n=S_{n+1}-S_n$ you should be able to get a first degree formula for $A_n$.

Now- Notice that $S_n-S_1=\sum_1^{n-1}A_n$, and since $S_1$ is easy to calculate, and $\sum A_n$ is not to tricky- this should give you the solution.

good luck!

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Sorry, I am in a bit of a hurry, but it should be this sequence.

To see this, just remark that you are summing up difference of consecutive squares (-1+4) + (-9+16) + ..., and these difference are the odd integers congruent to 3 mod 4 (i.e., 3, 7, 11...).

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You can separate this sum into two, one for $k$ odd and another for $k$ even, writing $k=2t+1$ and $k=2t$, respectively. You can also try finding $\sum k^2 z^k$ and then set $k=-1$.

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Yet another way (although not as simple as those already suggested) is to write it as $$-\sum_{k=1}^{2n} k^2 + 2 \sum_{m=1}^{n} (2m)^2$$ and use the known formula for a sum of squares.

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your formula looks good but how did you tranform my formula and get yours? – Martynas Feb 16 '11 at 14:48
@user7148: If O and E are the sums of the odd and even squares, respectively, then your sum is E-O, and I've written it as -(E+O)+2E. – Hans Lundmark Feb 16 '11 at 18:37

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