Serie-ous Series

Calculus Level 3

Which of the following series converge?

Series $\large \color{#D61F06}{A:}$ $\LARGE 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\ldots$

Series $\large \color{#3D99F6}{B:}$ $\LARGE 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots$

Series $\large \color{#20A900}{C:}$ $\LARGE 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\ldots$

Series $\large \color{#3D99F6}{D:}$ $\LARGE \frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\ldots$ (Reciprocal of Primes)

Series $\large \color{#D61F06}{E:}$ $\LARGE -\frac{1}{2}+\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{11}+\ldots$ (Reciprocal of Primes)

Series $\large \color{#20A900}{F:}$ $\LARGE \frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\ldots$ (Reciprocal of Fibonacci numbers)

Series $\large \color{#D61F06}{G:}$ $\LARGE 1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}-\ldots$

Series $\large \color{#3D99F6}{H:}$ $\Large \frac{1}{1}\Bigg(\frac{1}{1}-1\Bigg)+\frac{1}{2}\Bigg(\frac{1}{2}-1\Bigg)+\frac{1}{3}\Bigg(\frac{1}{3}-1\Bigg)+\ldots$

Series $\large \color{#20A900}{I:}$ $\Large \displaystyle \sum_{n=0}^\infty F_n \Bigg(\frac{2}{3}\Bigg)^n$

Series $\large \color{#3D99F6}{J:}$ $\Large 1-2+3-4+5-6+\ldots$

Note that $F_n$ is a Fibonacci number

Please do read the solution after attempting...

None of Them

\color{#D61F06}{A}\color{#3D99F6}{DH}\color{#20A900}{I}\color{#3D99F6}{J}

\color{#20A900}{C}\color{#D61F06}{E}\color{#20A900}{F}\color{#D61F06}{G}\color{#3D99F6}{J}

\color{#3D99F6}{B}\color{#20A900}{C}\color{#D61F06}{E}\color{#20A900}{F}\color{#D61F06}{G}

\color{#D61F06}{E}\color{#20A900}{F}\color{#D61F06}{G}\color{#20A900}{I}\color{#3D99F6}{J}

\color{#3D99F6}{B}\color{#20A900}{C}\color{#D61F06}{G}\color{#3D99F6}{H}\color{#20A900}{I}

All of Them

\color{#3D99F6}{D}\color{#D61F06}{EG}\color{#3D99F6}{H}\color{#20A900}{I}

1 solution

Digvijay Singh
Jun 4, 2015

Series $\large \color{#D61F06}{A:}$ Divergent because it is a harmonic series and any such series always diveges. More information here

Series $\large \color{#3D99F6}{B:}$ Convergent. Converges to $\large \ln2$ ; it can be shown using Mercator_series

Series $\large \color{#20A900}{C:}$ Convergent. Converges to $\large \frac{1}{e}$ ; it can be shown using the Taylor Series of $\large e^x$

Series $\large \color{#3D99F6}{D:}$ Divergent; as proved by Leonhard Euler in 1737.

Series $\large \color{#D61F06}{E:}$ Convergent. Click here for more information. Click here for the integer sequence of the resulting number.

Series $\large \color{#20A900}{F:}$ Convergent. Converges to Reciprocal Fibonacci constant $(\large ψ)$ . Click here and here for more information.

Series $\large \color{#D61F06}{G:}$ Convergent. Converges to $\large \frac{\pi^2}{12}$ . Click here for more information.

Series $\large \color{#3D99F6}{H:}$ Divergent; beacause it can be rearranged easily to $\large \zeta(2)-$ Series $\large \color{#D61F06}{A}$ and because Series $\large \color{#D61F06}{A}$ is divergent, the whole sum becomes divergent.

Series $\large \color{#20A900}{I:}$ Divergent; because the series of type $\large \displaystyle \sum_{n=0}^\infty F_n (x)^n$ converges only when $\large |x|<\frac{1}{φ}$ , and in this case, $\large \frac{2}{3}\approx 0.666...$ is greater than $\large \frac{1}{φ}\approx 0.61803$ . For more information, click here .

Series $\large \color{#3D99F6}{J:}$ Divergent; because its sequence of partial sums, $\large (1, -1, 2, -2, ...)$ , does not tend towards any finite limit. A paradoxical result to the sum was computed by Leonhard Euler, that is $\large 1-2+3-4+5-6+\ldots=\frac{1}{4}$ but this was later proved wrong. Click here for more information.

Serie-ous Series

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