Solving for a variable in a summation? This semester, I'm taking Calculus two. I'm working on a project for myself where I will need to solve the for a variable, inside a summation, such as "i" in the equation the one below:
$$
0=-20+\sum _{n=1}^{\infty \:}\:\frac{2}{\left(1+i\right)^n}
$$
Without telling me how to solve, is this
A) possible
B) something that would be explained in a Calc 2 course
C) a hint, please? :)
 A: 
Without telling me how to solve, is this A) possible

I imagine you mean to solve for $i$ in the summation? It would be possible, in theory: just find a closed form for the summation. A similar example would be the below (assume $|x|<1$):
$$0 = -20 + \sum_{n=1}^\infty x^n$$
Say you wanted to solve for $x$ here; well, a priori, it seems very difficult, because there's infinitely many $x$'s. However, recall the form for a geometric series:
$$\sum_{n=1}^\infty x^n = \frac{1}{1-x}$$
Then the equation is just
$$0 = -20 + \frac{1}{1-x}$$
for which solving for $x$ is almost trivial.


Without telling me how to solve, is this B) something that would be explained in a Calc 2 course

I don't know if it would necessarily be explained in a Calculus 2 course, no; I don't recall it coming up in mine at least. However, figuring out identities, properties, and closed forms for certain series is a pretty central topic to Calculus 1 and 2; what you want to do is just an application of said work, really.


Without telling me how to solve, C) a hint, please? :)

Assuming your series converges, you can factor out a $2$, and then notice from exponent rules that this holds:
$$\sum_{n=1}^\infty \frac{1}{(1+i)^n} = \sum_{n=1}^\infty \left( \frac{1}{1+i} \right)^n$$
Then consider utilizing the geometric series.
