# Serie-ous Series

Calculus Level 3

Which of the following series converge?

Series $$\large \color{red}{A:}$$ $$\LARGE 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\ldots$$

Series $$\large \color{blue}{B:}$$ $$\LARGE 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots$$

Series $$\large \color{green}{C:}$$ $$\LARGE 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\ldots$$

Series $$\large \color{blue}{D:}$$ $$\LARGE \frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\ldots$$ (Reciprocal of Primes)

Series $$\large \color{red}{E:}$$ $$\LARGE -\frac{1}{2}+\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{11}+\ldots$$ (Reciprocal of Primes)

Series $$\large \color{green}{F:}$$ $$\LARGE \frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\ldots$$ (Reciprocal of Fibonacci numbers)

Series $$\large \color{red}{G:}$$ $$\LARGE 1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}-\ldots$$

Series $$\large \color{blue}{H:}$$ $$\Large \frac{1}{1}\Bigg(\frac{1}{1}-1\Bigg)+\frac{1}{2}\Bigg(\frac{1}{2}-1\Bigg)+\frac{1}{3}\Bigg(\frac{1}{3}-1\Bigg)+\ldots$$

Series $$\large \color{green}{I:}$$ $$\Large \displaystyle \sum_{n=0}^\infty F_n \Bigg(\frac{2}{3}\Bigg)^n$$

Series $$\large \color{blue}{J:}$$ $$\Large 1-2+3-4+5-6+\ldots$$

Note that $$F_n$$ is a Fibonacci number