Finding value of $$1+\frac{1}{5}-\frac{1}{7}-\frac{1}{11}+\frac{1}{13}+\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+\cdots$$
Try: I have solved in using Integration
But i am trying to solver it without using integration
Witting above series as $$\sum^{\infty}_{n=0}(-1)^n\bigg[\frac{1}{6n+1}+\frac{1}{6n+5}\bigg]$$
Now i am trying to solve it using euler reflection formula
Could not find any clue
could some help me to solve it
Best Answer
Utilizing the observation done by Mohammad Zuhair Khan we can explicitly write down the series as
$$1+\sum_{n=1}^\infty(-1)^n\left[\frac1{6n+1}-\frac1{6n-1}\right]=1-2\sum_{n=1}^\infty \frac{(-1)^n}{36n^2-1}$$
The latter form can be rewritten such that we can apply a sum identity of the cosecant function. To be precise we will use the formula
Thus, lets rewrite the given series in the following way
$$\begin{align} 1-2\sum_{n=1}^\infty\frac{(-1)^n}{36n^2-1}&=1+\frac16\left[-2\frac16\sum_{n=1}^\infty \frac{(-1)^n}{n^2-\left(\frac16\right)^2}\right]\\ &=1+\frac16\left[\pi\csc\left(\frac\pi6\right)-6\right]\\ &=\frac\pi6\csc\left(\frac\pi6\right) \end{align}$$