# 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

Please do read the solution after attempting...