Always deducing a formula for a summation of series is quite exciting and my interest topic. I tried to get a formula or easy way to get summation of power series of natural numbers. I found a way to solve it. There could be many ways to solve this problem. I like this because I found it by my self.

Generic formula for summation of Series

According to Bernoulli’s theory (n+1) power of (t+1) can be written as following equations.

 

 

 

Summing all above equations will result in following conclusion

 

Now extracting formula for summation of series

 

Applying above formula to find out formula for Substituting “i”=n, t=2 and n as n.

My wife challenged me to prove it for . So I am adding this proof for her.

Summation of series can also be solved by using bernoulli’s numbers, invented by Bernoulli in 18 th century. Go to this link

http://numbers.computation.free.fr/Constants/Miscellaneous/bernoulli.html