Hello I am in year 11 and I am doing Spesh in the holidays , to get a head start.
I was stuck at this series question in the text book Ex 4E Q12
Find 1−x^2 +x^4 −x^6 +x^8 −···+x^2m ,where m is even.
Also please explain the answer.
Thanks
Hey, I haven't done spec so I'm not 100% on how they would expect you to find it but here are some thoughts that may help:
This is an alternating series (it's a series where we see it alternating b/w + and -)
This then leads to thinking, well how do they get that alternation? We can do this by having (-1)^n and then when n is an odd number it will be (-1)^n = -1 but if n is an even number (-1)^2 = 1. (When you apply this double check if you start at positive or negative, sometimes you might do n + 1 rather than n as the power.)
Let's move on from the signs for a moment, we can easily see that we are incrementing the power of x by 2 each time. Since we start at a_0 = 1, we know that the power is starting at 0. We can represent this information using x^2n
Let's put this together:
a_n is the sum of (-1)^n x^2n
we are told that m is even (so that gives us a plus for our last term)
Hope this helps!