# Sum of Sequence of Reciprocals of 3 n + 1 Alternating in Sign

## Theorem

\(\displaystyle \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {3 n + 1}\) | \(=\) | \(\displaystyle 1 - \frac 1 4 + \frac 1 7 - \frac 1 {10} + \frac 1 {13} - \cdots\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\pi \sqrt 3} 9 + \dfrac {\ln 2} 3\) |

## Proof

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.16$