Calculate the sum $\sum\limits_{n=1}^{100} n^{2}(\frac{1}{2})^n$

Question

I want to Calculate the sum $$\sum_{n=1}^{100} n^{2}(\frac{1}{2})^n$$ .No idea on how to begin. im just starting to learn some basic discrete math.Im not interested just solving this example but on a more general way on how to calculate sums by seeing how this one is solved

Answer 1

Hint:$$f(N,x):=\sum_{n=1}^Nx^n=x{1-x^{N}\over1-x}\\ \implies x{d\over dx}f(N,x)=\sum_{n=1}^Nnx^n\\ \implies x{d\over dx}\left(x{d\over dx}f(N,x)\right)=\sum_{n=1}^Nn^2x^n$$

Answer 2

Most sums cannot be summed explicitly. Typically, you might first look to see if it is a geometric series (sum is the same word) then it will have an explicit sum.

e.g $\sum_{n=0}^{N}x^n=\frac{1-x^{N+1}}{1-x}$

Your sum is not of this form. The next thing you might try is to differentiate this expression to create powers of n multiplying the $x^n$ term. With a bit of algebra and differentiating twice you can make this work for your sum.
If each term can be written as a difference of two terms in the same sequence you can also find the sum

i.e. $\sum_{n=0}^{N}a_n=\sum_{n=0}^{N}(b_{n+1}-b_n)=b_{N+1}-b_0$

After that you are struggling.
More often than not though you are not really interested in the value of any given finite sum - a computer will calculate it for you. More often than not you are interested in an estimate of an infinite sum or whether it has a sum at all (which a computer cannot, in general, deal with). Then, quite often, the sum in question is compared to the infinite geometric series or one of its derivatives.
In general I would suggest that you become very familiar with the geometric series and its derivatives. That will be very useful to you in the long run.