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.

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