<s>[INST] summation of series (odd and even case) Can anyone please answer this for me,it involves alternate signs which is different from normal summation formula. 
Q:Find the sum to n terms of the series $ 1^3 - 2^3 + 3^3 - 4^3 + \ldots -(n-2)^3 + (n-1)^3 - n^3 $.
 [/INST] The situation is a little different in the case $n$ even than in the case $n$ odd. We deal with even $n$, say $n=2k$, and leave the case $n$ odd to you.
Our sum is equal to 
$$\left(1^3+2^3+3^3+\cdots +(2k)^3\right) -2\left(2^3+4^3+6^3+\cdots+(2k)^3\right).$$
This is equal to
$$\left(1^3+2^3+3^3+\cdots +(2k)^3\right) -16\left(1^3+2^3+3^3+\cdots+k^3\right).$$
Now use the perhaps familiar fact that
$$1^3+2^3+3^3+\cdots +w^3=\left(\frac{w(w+1)}{2}\right)^2.$$
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