How to calculate sum of this? #sum_(n=1)^oo(-1)^n n(n-1)x^n#
2 Answers
See below.
Explanation:
Considering
but
Explanation:
We begin by writing out some of the coefficients:
The first thing we want to look at is the coefficients (the degree of
The coefficients inside this parenthesis can be recognized as the binomial series with a power of
We notice that the exponents of all the terms in the parenthesis are bigger by two compared to the series we just derived, so we must multiply
This means that our series is (when it converges) equal to:
Just to verify that we didn't make a mistake, we can quickly use the Binomial Series to compute a series for
We can describe this pattern like so:
Since the first term is just
which is the series we started with, verifying our result.
Now we just need to find out the convergence interval, to see when the series actually has a value. We can do this by looking at the convergence conditions for the binomial series and find that the series converges when