Correct approach to calculating sum of series Im a first semester at a local university - the course is computer science.
We were handed out a training assignment for math, which is not graded yet. However, im having difficulties - i'm basically straight out of school and have never worked with such topics yet.
The task is as follows: Evaluate $$\sum_{k\geqslant 0} \frac{(-3)^k+5}{4^k}$$
I dont want anyone here to do my "assignment" for me, i just want to know the general approach to solving such tasks as i have no idea where to even start.
 A: Hints: if $\;|x|<1\;$ , then
$$\sum_{n=0}^\infty x^k=\frac1{1-x}$$
and
$$\sum_{n=0}^\infty\frac{(-3)^n+5}{4^n}=\sum_{n=0}^\infty\left(-\frac34\right)^n+5\sum_{n=0}^\infty\left(\frac14\right)^n$$
A: Make use of things like:


*

*$\sum_k (a_k+b_k)=\sum_k a_k+\sum b_k$ and $\sum_k ca_k=c\sum_k a_k$ if the sums on RHS exist.

*$\sum_{k=0}^{\infty}r^k=\frac1{1-r}$ if $|r|<1$

A: In case what you are missing is the definition of the sum rather than how to calculate it, have a look at this previous question.  
Is there a formal definition of convergence of series?
It includes this example
$$\sum_{i=1}^\infty a_i = \lim_{n\to \infty}\sum_{i=0}^n{a_i}=L$$
The formal definition is: for all $\epsilon>0$ there exists an $N$ such that for all $n>N$, $$\left|L-\sum_{i=0}^n{a_i}\right|<\epsilon$$ 
You can regard  this as a game.  If I claim that the series sums to L then you may challenge me with any accuracy, $\epsilon > 0$ and I need to respond with an $N$ such that if you sum the first $N$ terms then I will achieve your accuracy. If I can prove that I can always achieve your accuracy then we say that L is the limit.  Challenging me with $\epsilon = 0$ is not allowed since I am not claiming that any partial sum will equal my target, just that it will get as close as you like by going far enough.  
